On the Area Relationship Between a Triangle and the Triangle Formed by Its Medians
The study of triangle geometry has long captivated mathematicians due to its inherent elegance and the deep relationships between different properties of a triangle. One such intriguing relationship involves the comparison between the area of a triangle and the area of a triangle formed by its medians. This result has far-reaching implications in various mathematical fields and continues to provide insights into geometric transformations and their properties.
The Median Triangle: Definition and Significance
In any given triangle, a median is a line segment that joins a vertex to the midpoint of the opposite side. A triangle, by definition, has three medians, and these medians are concurrent at a point called the centroid. This centroid divides each median into two parts, with the segment connecting the vertex to the centroid being twice the length of the segment connecting the centroid to the midpoint of the opposite side.
When the three medians of a triangle are used as the sides of a new triangle, the resulting triangle is known as the median triangle. While this geometric construction is simple, its relationship with the area of the original triangle reveals deeper insights into the triangle’s structure and properties.
Area Relationship Between the Original Triangle and the Median Triangle
A fascinating result in triangle geometry reveals that the area of the triangle formed by the medians is exactly 75% of the area of the original triangle. In mathematical terms, if represents the area of the original triangle and represents the area of the triangle formed by the medians, the following relationship holds:
S/s=4/3
This formula indicates that the area of the original triangle is times the area of the median triangle. This relationship arises from the geometric properties of the medians and their connection to the centroid.
Derivation of the Formula
To derive this area relationship, it is essential to recognize that the median triangle is similar to the original triangle. The medians divide the original triangle into smaller triangles, each of which is proportional to the original triangle. By applying principles of geometric similarity and proportionality, one can show that the area of the median triangle is of the area of the original triangle.
The factor comes from the scaling of the areas due to the centroid’s influence on the medians. The centroid acts as a point of balance, and it is through this balancing point that the areas of the two triangles are related in the manner described.
Applications and Importance
This area relationship has important applications in multiple areas of mathematics and physics. In geometry, it aids in understanding the properties of triangle transformations, while in optimization and design, it helps in problems where the centroid and medians play a role in determining structural properties.
Furthermore, this result enhances our understanding of how transformations, such as replacing the sides of a triangle with its medians, can affect area while preserving similarity. It also highlights the efficiency of using medians in various geometric calculations.
Conclusion
The relationship between the area of a triangle and the area of the triangle formed by its medians is a profound result in geometric analysis. The fact that the area of the median triangle is times that of the original triangle demonstrates the deep interconnections within the geometry of triangles. This result not only contributes to theoretical mathematics but also has practical implications in various fields where geometric transformations are employed.
At the end I lived in rented rooms. Desolate side streets. No elevator. Creaking steps. Paint chips in the water glass. Cockroaches in the bathtub. Bed by the wall. Dark convoluted mattress stains like an inkblot ghost. No hot water. Smell of old blood in the closet. Home for a week, home for a month. Then another city. Another room. Another name on the newspaper. Another set of identification letters for the television stations.
If he was in the South, I traveled south. When he ventured West, I followed west. The moonlight shines behind his fingers as he picks up the knife. The shadows unfold as I raise my hand. I wipe my forehead. I close my eyes.
I feel the wounds. I hear the screams.
Is this the room where the pregnant girl perspired during the hasty abortion that ruined the cheap bedspread? Is this the closet where the old watchman hanged himself, unable to hear the sound of his own voice? Maybe it is the place where the weary salesman raised the revolver to his temple. At that moment, a child sitting in a train on the elevated platform just beyond the salesman’s window put into his mouth a hard candy shaped like a bullet. Or could this be the last room for a killer? A deranged man? A monster unable to refrain from the dark urge, deliriously craving the final peace of his own destruction? Every room has a death story. Every room is another museum filled with the irremovable or unnoticed traces of someone’s fatal moments.
There was the vigorously applied razor blade left imbedded in the chunky soap bar. Dark flakes of hemoglobin were scattered across the white rectangle. They blew away as I raised the bathroom window with a bang. Three greasy fingerprints on the dull grey fuse box panel prefaced an outline of feet scorched on the shabby wood floor. Shards of a broken iodine bottle in the hallway leading to the toilet. Soiled grasp marks on the matrix of jaundiced damp sewage pipes. Nylons twisted into a noose lying like a coiled snake in a heap by the fire escape. An iridescent scabrous square of rat poison in the center of the loop. Crusts of rancid vomit in the Bible drawer. Maggots pinching through the Revelations.
A symbolic image, no doubt. The kind of thing that might appear in some controversial film about damnation, or the dissolution of religious belief. Dearest father, I did not forget your lessons. Everything I have seen throughout my life has been viewed within my own personal frame. Without really knowing why, the importance of a thing always depended on its visual content. I never understood the world, or its people, or its objects, unless I was making some kind of visual conclusion about the relationships between things. I could never resist what I must call a supreme demand, from somewhere within my nature, to establish and construe elaborate connections between all that my senses digested. It is as though my subconscious was engaged in some kind of esoteric archaeology, as though everything that could be depicted and suggested, especially all things that seemed destined to have a relationship, that somehow all this was already so, had been so, and now it was the duty of my mind’s divination to uncover what was, to reconstruct and display it, like a great structure or artifacts uncovered in a dig. It was as though my imagination had inherited some kind of perverted obligation from the teachings of my father, or perhaps my imperfect soul had made it perverse. Now I feel a great shame in all this, I can see the great reluctance that prevented me from true communion with others, yet I cannot deny the great understanding that depended on the power of the imagination, the interiority of consciousness, the relativity of perception and cognition. Did I unwittingly turn your wisdom into a comedy of errors, dear father? Did I somehow turn your spiritual warnings about the dangers of illusion into a rationale for the processes of illusion? I know you were genuine in your heart. You never gave me a stone when I asked you for bread. You never gave me a serpent when I asked you for a fish. Somehow the light of my body depended upon an evil eye, the false camera eye that filled my body with light that is darkness.
Shotgun blast blood outline, contours like a hologram fixed upon the wall after the trigger was pulled. Here the body remained too long, and there was too much heat, too little maid service. Gas mask swinging on the knob of the cellar door, hollow eyes sunken deep like a desert bone animal face. Cracked plastic tube of the hair blower in the empty stained fish tank once filled with water. Eyelashes brittle next to the coral house on the bottom, evidence of a successful electrocution long ago. Hysterical suicide confessions scrawled in lipstick across the large pages of the telephone book still in place atop the decrepit wooden stand by the lobby desk. Stench of the manager’s fingers as he flips through the book in search of a clean page. Monotony of his practiced gestures as he hands me the key, looks over the desk to be sure I have luggage, places the pen in the center of the decaying registration log, sits back on his stool, lighting another cigarette as he watches me ascend the stairs, wondering if I will become another suicide, another body carried out on the red rubber stretcher. A large cockroach does not escape the trained assault of his shoe. Its inner matter bursts with a gush as I turn the key to my room. Slowly the bent dusty blades of a fan turn about. The cockroach antennae twist a few times. I shut the door.
Peter J. Dellolio
The Confession is available here from Barnes and Noble.
Meet Samuel Beckett With Richard Wilson 2015 Manufacturing Intellect Princeton University Library Playing the Spectator While Waiting For Godot, Kimberly Bohman-Kalaja, The Princeton University Library Chronicle, Vol. 68, No. 1-2 Winter 2007, Princeton University Library Publishers.
Discuss the use of repetition and doubling as dramatic devices in Beckett’s Waiting for Godot. Or Bring out the significance of the stage setting in Waiting for Godot or in Look Back in Anger. Or Discuss the theatre of the absurd and connect it to some of its social and philosophical antecedents.
That postmodernist Irish tragicomical Waiting For Godot is a poetic drama of the Anglo-Franco absurdist tradition that evades both the meaning of life and purpose and that of memory and jurisdiction as envisioned by the vaudeville stock buffoon archetypal everyday humanity country bumpkins and fool-like jester tramps.
These protagonists Vladimir and Estragon’s histrionic rhetorics “Yesterday’s evening it was black and bare. Now it’s covered in leaves” and “It must be the spring” respectively delineate the trajectory of stage directions behind the stage and alleyways of a baffling generation of scholarly drama critics. Time is a patterning of memories in a narrative sequence as observable by these characters’ microcosmic natural world amidst blasted heaths and ruined countryside. Representations of recurrent imageries associated with boots and hats, gastric inflammation, and pouches of belching bear resemblance to outfit wardrobe and food crises prevalence of French resistance of the post world war epoch.
Emissary’s implication of Godot’s continual dismissal is lachrymose news to the readers of existentialism and nihilism. After all Pozzo’s declarative “Have you not done tormenting me with your accursed time?” postulates that the natural world is a purgatory without a promissory note of salvation as envisioned by these tramplike vagabonds; they cannot reminisce on past memories and are thus entwined within this gossamery of past and present spatiotemporality to be certain about who they are, where they are and why they are like rhetorical questions.
Estragon’s and Vladimir’s hanging upon the tree is a figurative trope of melodramatic hyperbolism that concerns finding meaning within a meaningless world. Lucky’s beastly burdensome stoicism [lifting of sand bags every now and then and then dragging them down to relift them] subjective to Pozzo’s tyrannical regime upon the behest of mindless and purposeless errand is symbolic of power dynamics concerning humanity’s enslavement to chasmic maze.
Lucky being deafened and Pozzo being blind incriminate subversion of power polity through the inversion of power dynamics, through banishment of colonial hegemony and thus proclaim emancipation to freedom by resistance and rebellion. That literature laureate absurdist and existentialist playwright Samuel Beckett crafts electrifying and spellbinding aural specks of allegorical enchantment in canonizing the fiction of absurdist poetic drama. After all, this is an allegory of the human condition for eternity as if we are cataclysmically falling with the rolling boulders from the cliff.
Fatalistically these tramp protagonists are eternalized for waiting and Beckett has transformed the destitution of mankind into exaltation through Lucky’s personae: “He’s Lucky to have no more expectations.” Furthermore, the polar binaries between the powerful Pozzo and the powerless Lucky, Estragon, and Vladimir insinuate extended metaphors of the Cold War, the French Resistance, and the Irish rebellious spirits of the nationalist freedom movement.
“Let us not waste our time in idle discourse! Let us do something while we have a chance.” Vladimir’s speech is evoked in implication of salvaging the quagmire of Pozzo and Lucky’s funebrial crisis. Angst and pangst of existentialist crisis has been translated to the traumatic psyche of these priggish and prudential beings. However the stage directions of being stationary connotes their dwarfish dormancy and alienated stagnation. That the pointlessness of existence is implicated in salvation being awaited by external force and that self is incapable of self-knowledge. In cloak and dagger connotations of Estragon and Vladimir symbolically represents ego and id while Pozzo and Lucky symbolically represents superego.
As a result these characters are alter egos or shadows or persona soul image of themselves weaved by the gossamery of existentialist crisis. In this context, Lucky is the shadow of the superego of the egocentric Pozzo whose emotion becomes repressed pouring forth of the unconscious state through monologue.
Estragon is feminized with sensitive, irrational and poetic traits while Vladimir is masculinized with rational, contemplative and intellectual traits. Godot is a political satirical idiom of modern popular culture symbolic of the gothic monsterish figure of loathsome whangdoodle as dracula macabre. Pathos of nothingness is a dire catharsis by the crucial existentialists’ plight engendering from being sublime to travesty within universalistic spatiotemporality by the indication of “A country road”. “A tree”. “Evening”.
Domineering colonizer master Pozzo with his whip and the subservient colonized subaltern Lucky’s servility in burdensome stoical endurance is the inversion of the amnesty between Estragon and Vladimir despite these individualists’ nihilistic despair with insurmountable frustrations. Antiphrasis of stage directions hint to “They do not move” despite speech acts of voluntary action: “Let’s us go” furthermore metaphorically suggestive of philosophical pessimism as embodied silence, stasis, absence and negation.
Becket’s poignant revelatory envisioning from Biblical allusions point out that “Do not despair, one of the thieves was saved; do not presume, one of the thieves was damned.” Although the tragicomedy lacks female reproductive machinery however, the tree is symbolic of that utopian hope in a world of futility.
Frugal and mundane existence in a characteristic bleak landscape in expectation and anticipation of the messianic Saviour Christ through the mediation of the emissarial convoy exhibit the maudlin encumbrance of these stock characters like vaudeville fools and country bumpkins in mainstream absurdist realism. “I’ll never forget this carrot. The more you eat, the worst it gets. I’ll get used to the muck as I go along.”
These dialects are philosophical prompts propounded by the childish, materialistic, feminist, poetic, melodramatic Estragon and rational intellectualist wimpy guffaw of Vladimir contrasting differences of their outlook in life. The essence of struggling and wriggling is both bogus and vague as contemplated by these speculative skeptical states of affairs. Godot might be a satirical human condition of both waiting and achievement throughout Christmas, birthday celebration, job prospect, love of the life, funeral anniversary and so forth. Sadomasochism of Pozzo and Lucky are allegorically satirized by brevity of intertextual allusions that mirrors habitual distraction and interruption that embodies Didi and Gogo’s world of nihilistic pessimism, stasis and repetition, skepticism and ambiguity.
Their forlorn and obscuring of train of thoughts and chain of events, forgotten memories, obliviousness of dreams, discarding of dialogues and abandonment of suicide attempts are verily brought to the foray of this justification. Language has lost the essence of the core of communication by the farrago of charlatantry and buffoonery in Lucky’s monologue. Audiences would walk out by the off stage characters’ frustration and oppression after all in correspondence with the effect of defamiliarization. Lucky isolated island of retreat from dialogism critiques the purgatorial nightmare pestering into the infested microcosmic existence of these slapstick vaudeville country bumpkins tramps. Lucky is the symbolic thinktank Beckettian institution which dismantles establishment of linguistic games and sheds light on the furthering of ideas into the dialogic proximity.
After being traumatized and tortured by these existentialist characters, Lucky is doomed into thinking and functioning as Pozzo’s porter.
Further References Youtube Podcasts and Documentary Films and Lecture Presentations Seminary Samuel Beckett’s Waiting for Godot, Professor Dr. Nick Mount, Department of English, University of Toronto The Meaning of Godot, Professor Dr. David Pattie, Department of Drama and Theatre, University of Birmingham Theatre and Language: Samuel Beckett’s Waiting for Godot, Professor Dr. Belinda Jack, Gresham Professor of Rhetoric, Gresham College, London, UK Cambridge PhDcasts John Gallagher presents Any Wimbush’s Samuel Beckett and Quietism Ian McKellen Discusses “Waiting For Godot” Staging Shakespeare
“I didn’t even know his name,” she whispered softly. She looked at me. “Do you know it?” she asked. She had bright green eyes.
“Johnson,” I said gruffly.
She nodded.
“Can you tell me how it happened?” I asked.
“I met him on the bus. We talked. He seemed nice.”
I waited.
“He said, do you want to get a coffee, so we could talk some more.”
“You got off at the stop on Rogers?”
She nodded, but said no more.
“What happened next?”
“We got our coffees at the McDonalds and then strolled to the park. We talked for a while and then I looked around and we were suddenly in a woods. He…grabbed my arm and twisted it. I tried to yell but he put his hand over my mouth. He started to rip my clothes off me.”
“You’re doing good,” I told her. “How did it end?”
“He had me on the ground and was on top of me and I opened my purse with my free hand and pulled out my pistol and stuck it in his belly and pulled the trigger three times…”
I waited a minute. “And then?” I prompted.
“And then the police came. Someone in the park must have heard the shots and called them.”
I held up a transparent evidence bag with a Glock inside. “Is this your gun, Caroline?” I asked gently.
She nodded.
“You’re the legal owner of this weapon?” I asked. I had already checked the registration and the data bases. She was legal.
She nodded again.
“How did you happen to be carrying it?”
“Habit. I always carry it with me, everywhere, ever since the first time.”
