2019 AIME II Problems/Problem 1
Problem
Two different points, and , lie on the same side of line so that and are congruent with , , and . The intersection of these two triangular regions has area , where and are relatively prime positive integers. Find .
Solution
[asy] size(10cm); pair A=(0,0), B=(9,0), C=(15,8), D=(-6,8), C_1=(15,0), P=(9/2,12/5); draw(B--C--D--A); draw(C--P--D); draw(B--C_1--C,dashed); draw((A+B)/2--(C+D)/2,dashed); label("",A,SW); label("",B,S); label("",C_1,SE); label("",C,NE); label("",D,NW); label("",P+(0.5,0),E); draw(A--B--P--cycle,red+linewidth(1.1)); dot(A); dot(B); dot(C); dot(D); dot(P); dot(C_1); label("",(A+B)/2,dir(A--B)*dir(-90)); label("",(B+C)/2,dir(B-C)*dir(90)); label("",(B+C_1)/2,dir(B-C_1)*dir(90)); label("",(C+C_1)/2,dir(C_1-C)*dir(90)); label("",(C+D)/2,dir(C--D)*dir(-90)); label("",(2*P+A+B)/4+(-0.1,-0.3),E); [/asy]
Note that and so if is the projection of onto then . It follows that . Let the diagonals and intersect at , then with similarity factor . Thus the height of is so and hence the answer is as desired.