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GeometryQuestion and Answers: Page 103 |
Let A′, B′ and C′ be points on the sides BC, CA and AB of the triangle ABC. Prove that the circumcircles of the triangles AB′C′, BA′C′ and CA′B′ have a common point. Prove that the property holds even if the points A′, B′ and C′ are collinear. |
Let ABCD be a convex quadrilateral. Prove that AB.CD + AD.BC = AC.BD if and only if ABCD is cyclic (Ptolemy′s theorem). |
In the interior of a quadrilateral ABCD, consider a variable point P. Prove that if the sum of distances from P to the sides is constant, then ABCD is a parallelogram. |
Let ABCD be a convex quadrilateral and let E and F be the points of intersections of the lines AB, CD and AD, BC, respectively. Prove that the midpoints of the segments AC, BD, and EF are collinear. |
Let d, d′ be two nonparallel lines in the plane and let k > 0. Find the locus of points, the sum of whose distances to d and d′ is equal to k. |
Let ABCD be a convex quadrilateral and let k > 0 be a real number. Find the locus of points M in its interior such that [MAB] + 2[MCD] = k. |
Let ABCD be a convex quadrilateral. Find the locus of points M in its interior such that [MAB] = 2[MCD]. |
Let ABCD be a convex quadrilateral and M a point in its interior such that [MAB] = [MBC] = [MCD] = [MDA]. Prove that one of the diagonals of ABCD passes through the midpoint of the other diagonal. |
number of positive integers a and b and c satisfying a^b^c b^c^a c^a^b =5abc |
A cirlce is drawn to touch the sides of a triangle whose sides are 12cm,10cm,and 9cm. Find the radius of the circle. |
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multiply 3x+4y+5x−8y |
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3+5x=58 |
solve the equation: {5^x } +5x=140 please show workings..... |
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Pg 98 Pg 99 Pg 100 Pg 101 Pg 102 Pg 103 Pg 104 Pg 105 Pg 106 Pg 107 |