RMO and Pre RMO Geometry Tutorial Worksheet 1: Based on Geometric Refresher

1) Show that quadrilateral ABCD can be inscribed in a circle iff $\angle B$ and $\angle D$ are supplementary.

2) Prove that a parallelogram having perpendicular diagonals is a rhombus.

3) Prove that a parallelogram with equal diagonals is a rectangle.

4) Show that the diagonals of an isosceles trapezoid are equal.

5) A straight line cuts two concentric circles in points A, B, C and D in that order. AE and BF are parallel chords, one in each circle. If CG is perpendicular to BF and DH is perpendicular to AE, prove that $GF = HE$ .

6) Construct triangle ABC, given angle A, side AC and the radius r of the inscribed circle. Justify your construction.

7) Let a triangle ABC be right angled at C. The internal bisectors of angle A and angle B meet BC and CA at P and Q respectively. M and N are the feet of the perpendiculars from P and Q to AB. Find angle MCN.

8) Three circles $C_{1}, C_{2}, C_{3}$ with radii $r_{1}, r_{2}, r_{3}$ , with $r_{1}<r_{2}<r_{3}$ . They are placed such that $C_{2}$ lies to the right of $C_{1}$ and touches it externally; $C_{3}$ lies to the right of $C_{2}$ and touches it externally. Further, there exist two straight lines each of which is a direct common tangent simultaneously to all the three circles. Find $r_{2}$ in terms of $r_{1}$ and $r_{3}$ .