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MATH / In triangle ABC, AB=13, AC=15, and BC=14. Let I be the incenter. The incircle of triangle ABC touches sides BC, AC, and AB at D, E, and F, respectively. Find the area of quadrilate…

Problem

Answer published by the source. Consult the official source to check your work against its answer.

problem

In triangle ABCABC, AB=13AB = 13, AC=15AC = 15, and BC=14BC = 14. Let II be the incenter. The incircle of triangle ABCABC touches sides BCBC, ACAC, and ABAB at DD, EE, and FF, respectively. Find the area of quadrilateral AEIFAEIF. [asy] import geometry; unitsize(2 cm); pair A, B, C, D, E, F, I; A = (1,2); B = (0,0); C = (3,0); I = incenter(A,B,C); D = (I + reflect(B,C)*(I))/2; E = (I + reflect(C,A)*(I))/2; F = (I + reflect(A,B)*(I))/2; draw(A--B--C--cycle); draw(incircle(A,B,C)); draw(E--I--F); label("AA", A, N); label("BB", B, SW); label("CC", C, SE); label("DD", D, S); label("EE", E, NE); label("FF", F, NW); label("II", I, S); [/asy]
Plain-text mathematical notation (without MathML)
In triangle ABC, AB=13, AC=15, and BC=14.  Let I be the incenter.  The incircle of triangle ABC touches sides BC, AC, and AB at D, E, and F, respectively.  Find the area of quadrilateral AEIF.

[asy]
import geometry;

unitsize(2 cm);

pair A, B, C, D, E, F, I;

A = (1,2);
B = (0,0);
C = (3,0);
I = incenter(A,B,C);
D = (I + reflect(B,C)*(I))/2;
E = (I + reflect(C,A)*(I))/2;
F = (I + reflect(A,B)*(I))/2;

draw(A--B--C--cycle);
draw(incircle(A,B,C));
draw(E--I--F);

label("A", A, N);
label("B", B, SW);
label("C", C, SE);
label("D", D, S);
label("E", E, NE);
label("F", F, NW);
label("I", I, S);
[/asy]
Original LaTeX notation
In triangle $ABC$, $AB = 13$, $AC = 15$, and $BC = 14$.  Let $I$ be the incenter.  The incircle of triangle $ABC$ touches sides $BC$, $AC$, and $AB$ at $D$, $E$, and $F$, respectively.  Find the area of quadrilateral $AEIF$.

[asy]
import geometry;

unitsize(2 cm);

pair A, B, C, D, E, F, I;

A = (1,2);
B = (0,0);
C = (3,0);
I = incenter(A,B,C);
D = (I + reflect(B,C)*(I))/2;
E = (I + reflect(C,A)*(I))/2;
F = (I + reflect(A,B)*(I))/2;

draw(A--B--C--cycle);
draw(incircle(A,B,C));
draw(E--I--F);

label("$A$", A, N);
label("$B$", B, SW);
label("$C$", C, SE);
label("$D$", D, S);
label("$E$", E, NE);
label("$F$", F, NW);
label("$I$", I, S);
[/asy]

level

Level 5

type

Geometry

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Source and history

task_id
26c9fa97-f1fc-5b9c-84d2-cdca81649f59
task_key
geometry--test--26c9fa97-f1fc-5b9c-84d2-cdca81649f59
task_revision_id
1
config / split
geometry / test

Official source

initial import