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Question Number 16214 Answers: 2 Comments: 4

In ΔABC, r_1 , r_2 and r_3 are the exradii as shown. Prove that r_1 = (Δ/(s − a)) , r_2 = (Δ/(s − b)) and r_3 = (Δ/(s − c)) . Here s = ((a + b + c)/2) .

$In Δ A B C, r_{1}, r_{2} and r_{3} are the exradii$ $as shown . Prove that r_{1} = \frac{Δ}{s - a},$ $r_{2} = \frac{Δ}{s - b} and r_{3} = \frac{Δ}{s - c} . Here$ $s = \frac{a + b + c}{2} .$

Question Number 16194 Answers: 0 Comments: 21

Question Number 16140 Answers: 2 Comments: 0

Question Number 16110 Answers: 0 Comments: 1

let a_1 >a_2 >0 and a_(n+1) =(√(a_n a_(n−1 ) )) where n is greater than equal to 2 Then The sequence {a_(2n) } is (1) monotonic increasing (2)monotonic decreasing (3)non monotonic (4)unbounded

$let a_{1} > a_{2} > 0 and a_{n + 1} = \sqrt{a_{n} a_{n - 1}}$ $where n is greater than equal to 2$ $Then$ $The sequence {a_{2 n}} is$ $(1) monotonic increasing$ $(2) monotonic decreasing$ $(3) non monotonic$ $(4) unbounded$

Question Number 16108 Answers: 1 Comments: 1

Question Number 16077 Answers: 0 Comments: 0

Let ABCDE be an equiangular pentagon whose side lengths are rational numbers. Prove that the pentagon is regular.

$Let A B C D E be an equiangular$ $pentagon whose side lengths are$ $rational numbers . Prove that the$ $pentagon is regular .$

Question Number 16075 Answers: 0 Comments: 0

Prove that the perpendiculars dropped from the midpoints of the sides of a cyclic quadrilateral to the opposite sides are concurrent.

$Prove that the perpendiculars$ $dropped from the midpoints of the sides$ $of a cyclic quadrilateral to the opposite$ $sides are concurrent .$

Question Number 16074 Answers: 0 Comments: 0

Let K, L, M and N be the midpoints of the sides AB, BC, CD and DA, respectively, of a cyclic quadrilateral ABCD. Prove that the orthocenters of the triangles AKN, BKL, CLM and DMN are the vertices of a parallelogram.

$Let K, L, M and N be the midpoints of$ $the sides A B, B C, C D and D A,$ $respectively, of a cyclic quadrilateral$ $A B C D . Prove that the orthocenters$ $of the triangles A K N, B K L, C L M and$ $D M N are the vertices of a$ $parallelogram .$

Question Number 16072 Answers: 1 Comments: 3

Let ABCD be a convex quadrilateral. Prove that the orthocenters of the triangles ABC, BCD, CDA and DAB are the vertices of a quadrilateral congruent to ABCD and prove that the centroids of the same triangles are the vertices of a cyclic quadrilateral.

$Let A B C D be a convex quadrilateral .$ $Prove that the orthocenters of the$ $triangles A B C, B C D, C D A and D A B$ $are the vertices of a quadrilateral$ $congruent to A B C D and prove that the$ $centroids of the same triangles are the$ $vertices of a cyclic quadrilateral .$

Question Number 16071 Answers: 1 Comments: 0

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.

$Let A^{'}, B^{'} and C^{'} be points on the sides$ $B C, C A and A B of the triangle A B C .$ $Prove that the circumcircles of the$ $triangles {A B}^{'} C^{'}, {B A}^{'} C^{'} and {C A}^{'} B^{'}$ $have a common point . Prove that the$ $property holds even if the points A^{'},$ $B^{'} and C^{'} are collinear .$

Question Number 16070 Answers: 0 Comments: 0

Let ABCD be a convex quadrilateral. Prove that AB.CD + AD.BC = AC.BD if and only if ABCD is cyclic (Ptolemy′s theorem).

$Let A B C D be a convex quadrilateral .$ $Prove that$ $A B . C D + A D . B C = A C . B D$ $if and only if A B C D is cyclic ({Ptolemy}^{'} s$ $theorem) .$

Question Number 16069 Answers: 0 Comments: 0

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.

$In the interior of a quadrilateral$ $A B C D, consider a variable point P .$ $Prove that if the sum of distances from$ $P to the sides is constant, then A B C D$ $is a parallelogram .$

Question Number 16068 Answers: 2 Comments: 0

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.

$Let A B C D be a convex quadrilateral$ $and let E and F be the points of$ $intersections of the lines A B, C D and$ $A D, B C, respectively . Prove that the$ $midpoints of the segments A C, B D,$ $and E F are collinear .$

Question Number 16067 Answers: 1 Comments: 8

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.

$Let d, d^{'} be two nonparallel lines in the$ $plane and let k > 0 . Find the locus of$ $points, the sum of whose distances to$ $d and d^{'} is equal to k .$

Question Number 16066 Answers: 2 Comments: 0

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.

$Let A B C D be a convex quadrilateral$ $and let k > 0 be a real number . Find$ $the locus of points M in its interior$ $such that$ $[M A B] + 2 [M C D] = k .$

Question Number 16065 Answers: 0 Comments: 0

Let ABCD be a convex quadrilateral. Find the locus of points M in its interior such that [MAB] = 2[MCD].

$Let A B C D be a convex quadrilateral .$ $Find the locus of points M in its interior$ $such that [M A B] = 2 [M C D] .$

Question Number 16064 Answers: 0 Comments: 8

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.

$Let A B C D be a convex quadrilateral$ $and M a point in its interior such that$ $[M A B] = [M B C] = [M C D] = [M D A] .$ $Prove that one of the diagonals of$ $A B C D passes through the midpoint of$ $the other diagonal .$