New Zealand, Camp Selection 2018, problem 4
Problem. Let be a point inside triangle such that and . Let and be the midpoints of and respectively. Prove that .
Origin: New Zealand, Camp Selection 2018, problem 4.
Solution. Let's consider a homothety with center and scale factor . This homothety maps , , , where is the point on the ray such that .
Notice that since , lies on the circle with diameter and center . Thus, . These two segments, and , map to the segments and respectively under the homothety considered above. Therefore, . This, in turn, implies that . When we combine this with the condition , we obtain . This implies that the the quadrilateral is inscribed (cyclic). Therefore, . Finally, notice that under the homothety considered above the angle maps to the angle , thus , and we are done.