Geometry problem

Question

Let ∆ABC be a triangle with AB = 65, BC = 70, and CA = 75. A semicircle Γ with diameter BC is constructed outside the triangle. Suppose there exists a circle ω tangent to AB and AC and furthermore internally tangent to Γ at a point X. The length AX can be written in the form m√n (m root n) where m and n are positive integers that are not divisible by the square of any prime. Find m + n.

Answer 1

By setting up 3 equations with 3 unknowns and using

Wolfram Alpha to solve that system, the answer is confirmed.

Let z = the x-coordinate of the center of the circle that must lie on the angle bisector of angle A. The y coordinate of that center is 4/7 z.

The equation of the semi-circle (x-54)^2+(y-28)^2 = 35^2 (need additional restrictions to ensure top half of semi-circle). The equation of the circle that is internally tangent to the semi-circle: (x - z)^2 + (y- 4/7 z)^2 = (4/7 z)^2.

The equation of the line that passes through the 2 centers and therefore through the point of tangency:

y = (4/7 z - 28)/(z-54) [x-54] + 28, where the expression in front of ([x - 54] is the slope of the line through the centers.

Note that trigonometry was used to find the measure of angle A, (Law of Cosines) and the point that serves as the incenter of the triangle for the circle that is tangent to two sides lies on the angle bisector of angle A. Also to find the coordinates of point B, the equations of two circles with centers at A and C respectively along with relevant radii were solved for point of intersection.

The following Desmos graph depicts the diagram along with the results from Wolfram Alpha -- I gave up trying to solve the system manually. There might be other ways to solve this problem including finding the equation of the tangent line at the point of tangency and using the fact that the slope of that tangent line is the negative reciprocal of the slope of the line passing through the centers of the two circles.

The answers shown from Wolfram Alpha are based on the fact that the circles could be externally tangent and that the circle on side BC is not restricted to semi-circle.

desmos.com/calculator/zh3cg1gspk

Very interesting problem.