2008 IMO Problems/Problem 6
Let be a convex quadrilateral with . Denote the incircles of triangles and by and respectively. Suppose that there exists a circle tangent to ray beyond and to the ray beyond , which is also tangent to the lines and . Prove that the common external tangents to and intersect on
This problem needs a solution. If you have a solution for it, please help us out by.
Here are some hints:
Let B be the top vertex of triangle ABC, O and K are the centers of the incircles of triangles ABC and ADC with radii R and r, respectively. S is the center of the circumcircle tangential to the extensions of AB, AC and DC. And let
E be the foot of the projection of O to AB. U be the foot of the projection of O to AC. V be the foot of the projection of K to AC. M be the foot of the projection of S to DC. L be the foot of the projection of K to DC. L be the intercept of DC and AB.
/_EOB = /_LSC /_AOU = /_SOC /_ASO = /_KSC /_ASK = /_OSC /_LSB = /_KCO UV = BC – AB AU = VC OA. AK. cos(/_OAK) = OC. KC. cos(/_OCK) OK**2 = (R + r)**2 + UV**2 SK**2 = (R' + r)**2 + ML**2
Use sin (90-x) = cos x and cos(90-x) = sinx and characteristic of triangle a**2 = b**2 + c**2 - 2.b.c.cosine(angle) to solve.
Vo Duc Dien
|2008 IMO (Problems) • Resources|
|1 • 2 • 3 • 4 • 5 • 6||Followed by|
|All IMO Problems and Solutions|