1. The incircle of LABC is tangent to sides BC, AC, and AB at X, Y, and Z, respectively.
(a) Prove that AX,BY, and CZ are concurrent. (b) Given that BY = CZ, prove that LABC is isosceles. 2. Trapezoid ABCD has two right angles at A and D, an inscribed circle, and bases AB = a and CD = b. Find the area of ABCD in terms of a and b.
3. ABC is a 3-4-5 triangle and F is on AC.
(a) Find the length of segment BF if it divides triangle /ABC into two triangle of equal area. (b) Find the length of the shortest segment DE that divides triangle LABC into two regions of equal area.
4. In trapezoid ABCD, AB ll CD, AB = 6, BC = 4, CD = 1, and DA = 3. Prove that AD IBC.
5. Segment AF is an altitude in triangle LABC with AB = 30, AC = 40, BF = 18, and FC = 32.
(a) Find the distance between the circumcenter and the incenter of triangle LABC. (b) Find the distance between the incenters of triangle IABC and triangle /AFC.
(c) If AF is the altitude of right triangle ABC, with LA = 90°, r is the inradius of triangle bABC, s is the inradius of triangle AABF, and t is the inradius of triangle AAFC, prove that r+s+t = AF. (d) If AF is the altitude of right triangle AABC with LA = 90°, r is the inradius of triangle AABC, s is the inradius of triangle LIABF, and t the inradius of AAFC, prove that r2 = s2 + e. 6. Cevian AQ of equilateral triangle ABC is extended to meet its circum-cirIce at P. Let AP = d, BP = e, CP = f, and AB = s. Prove that
(a) d = e+ f (b) d2 + e2 + f2 = 2s2. 7. On a circle of diameter AB let C be the midpoint of one of the arcs AB and let P be an arbitrary point on the other arc AB. Prove that (PA+ PB)2 = 2PC2.
8. An isosceles trapezoid with side-lengths 2, 3, 2, and 4 is inscribed in a circle. Find the radius of the circle and justify your answer. 9. ABCD is a cyclic quadrilateral with perpendicular diagonals intersect-ing at E. (a) If AB = a, BC = b, CD = c, and DA = d, prove that + K ADCD ac 2 bd (b) If R is the radius of the circumcircle, prove that EA2 + EB2 + EC2 + ED2 = 4R2. 10. Prove that in a cyclic quadrilateral ABCD with perpendicular diago-nals, the distance from the circumcircle to a side is half the length of the opposite side. 11. Quadrilateral ABCD is inscribed in a semicircle of diameter AD = 2. If AB = x, BC = y, and CD = z, prove that z2 + y2 + z2 + xyz = 4.