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MATH / In triangle ABC, ∠BAC=72^(∘). The incircle of triangle ABC touches sides BC, AC, and AB at D, E, and F, respectively. Find ∠EDF, in degrees. [asy] import geometry; unitsize(2 cm); …

Problem

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

problem

In triangle ABCABC, ∠BAC=72∘\angle BAC = 72^\circ. The incircle of triangle ABCABC touches sides BCBC, ACAC, and ABAB at DD, EE, and FF, respectively. Find ∠EDF\angle EDF, in degrees. [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(F--D--E); label("AA", A, N); label("BB", B, SW); label("CC", C, SE); label("DD", D, S); label("EE", E, NE); label("FF", F, NW); [/asy]
Plain-text mathematical notation (without MathML)
In triangle ABC, ∠BAC=72^(∘).  The incircle of triangle ABC touches sides BC, AC, and AB at D, E, and F, respectively.  Find ∠EDF, in degrees.

[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(F--D--E);

label("A", A, N);
label("B", B, SW);
label("C", C, SE);
label("D", D, S);
label("E", E, NE);
label("F", F, NW);
[/asy]
Original LaTeX notation
In triangle $ABC$, $\angle BAC = 72^\circ$.  The incircle of triangle $ABC$ touches sides $BC$, $AC$, and $AB$ at $D$, $E$, and $F$, respectively.  Find $\angle EDF$, in degrees.

[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(F--D--E);

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

level

Level 5

type

Geometry

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Source and history

task_id
68f068fb-a578-52f5-b2d7-a39c1dee2b28
task_key
geometry--test--68f068fb-a578-52f5-b2d7-a39c1dee2b28
task_revision_id
1
config / split
geometry / test

Official source

initial import