Problems in Plane and Solid Geometry v.1 Plane Geometry by V. Prasolov D.Leites (translator)

By V. Prasolov D.Leites (translator)

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Extra info for Problems in Plane and Solid Geometry v.1 Plane Geometry

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Denote the intersection points of chords M C and M D with chord AB by E and K. Prove that KECD is an inscribed quadrilateral. 15. Concider an equilateral triangle. A circle with the radius equal to the triangle’s height rolls along a side of the triangle. Prove that the angle measure of the arc cut off the circle by the sides of the triangle is always equal to 60◦ . 16. The diagonals of an isosceles trapezoid ABCD with lateral side AB intersect at point P . Prove that the center O of the inscribed circle lies on the inscribed circle of triangle AP B.

The inscribed circle of triangle ABC is tangent to sides AB and AC at points M and N , respectively. Let P be the intersection point of line M N with the bisector (or its extension) of angle ∠B. Prove that: a) ∠BP C = 90◦ ; b) SABP : SABC = 1 : 2. 42. Inside quadrilateral ABCD a point M is taken so that ABM D is a parallelogram. Prove that if ∠CBM = ∠CDM , then ∠ACD = ∠BCM . 43. Lines AP , BP and CP intersect the circumscribed circle of triangle ABC at points A1 , B1 and C1 , respectively. On lines BC, CA and AB points A2 , B2 and C2 , respectively, are taken so that ∠(P A2 , BC) = ∠(P B2 , CA) = ∠(P C2 , AB).

Circles S1 and S2 intersect at points A and B. Line M N is tangent to circle S1 at point M and to S2 at point N . Let A be the intersection point of the circles, which is more distant from line M N . Prove that ∠O1 AO2 = 2∠M AN . 101. Given quadrilateral ABCD inscribed in a circle and such that AB = BC, prove that SABCD = 21 (DA + CD) · hb , where hb is the height of triangle ABD dropped from vertex B. 102. Quadrilateral ABCD is an inscribed one and AC is the bisector of angle ∠DAB. Prove that AC · BD = AD · DC + AB · BC.

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