Triangle ABC has sides AB = 8, BC = 7, and AC = 9. The segments AB and AC are extended and a circle O is constructed exterior to the triangle and is tangent to BC at D and the extended lines through AC and AB at N and M, as shown. What is the radius of circle O?
Let ABC be the triangle with AB = 1, AC = 3, and ∠BAC = π/2. If a circle of radius r > 0 touches the sides AB, AC and also touches internally the circumcircle of the triangle ABC, then the value of r is ___.
Triangle ABC has a right angle at B. Let Q be along BC and P be along AB such that AQ bisects angle A and CP bisects angle C. If AQ = 9 and CP = 8√2, what is the length of the hypotenuse AC?
Triangle ABC has AB = BC = 10 and AC = 10√2. The triangle is folded so that point A falls on the midpoint M of BC. The crease endpoints P, Q are along sides AB and AC, respectively. What is the length of the crease PQ?
A triangle with sides AB = 3, BC = 4, and AC = 5 has an inscribed square PQRS with side QR along the side AC. What is the area of the square?
