2021 Fall AMC 10A Problems/Problem 15

Isosceles triangle has , and a circle with radius is tangent to line at and to line at . What is the area of the circle that passes through vertices , , and

Solution 1

Let the center of the first circle be By Pythagorean Theorem, Now, notice that since is degrees, so arc is degrees and is the diameter. Thus, the radius is so the area is

- kante314

Solution 2 (Similar Triangles)

import olympiad;
unitsize(50);
pair A,B,C,D,E,I,F,G,O;
A=origin; B=(2,3); C=(-2,3); D=(4.3,6.3); E=(-4.3,6.3); F=(1,1.5); G=(-1,1.5);
O=circumcenter(A,B,C); // olympiad - circumcenter
I=incenter(A,D,E);
draw(A--B--C--cycle);
dot(O);
dot(I);
dot(F);
dot(G);
draw(circumcircle(A,B,C)); // olympiad - circumcircle
draw(incircle(A,D,E));
draw(I--B);
draw(I--C);
draw(I--A);
draw(rightanglemark(A,C,I)); 
draw(rightanglemark(A,B,I));
draw(O--F);
draw(O--G);
draw(rightanglemark(A,F,O)); 
draw(rightanglemark(A,G,O));


label("$O$",O,W);
label("$A$",A,S);
label("$B$",B,N);
label("$C$",C,W);
label("$D$",F,S);
label("$E$",G,W);

label("$3\sqrt{6}$",(1.5,1.5),S);
label("$3\sqrt{6}$",(-1.5,1.5),S);
label("$5\sqrt{2}$",(1,3.625),N);
label("$5\sqrt{2}$",(-1,3.625),N);
label("$I$",I,N);
label("$r$",(-0.25,1.5),E);
label("$r$",(0.5,2.125),S);
add(pathticks(A--F,1,0.5,0,2);
add(pathticks(F--B,1,0.5,0,2); 
add(pathticks(A--G,1,0.5,0,2);
add(pathticks(G--C,1,0.5,0,2);



 (Error making remote request. Unknown error_msg)

Because circle is tangent to at . Because O is the circumcenter of is the perpendicular bisector of .

Solution in Progress

~KingRavi

See Also

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.