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GeometryQuestion and Answers: Page 103

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.

LetA,BandCbepointsonthesidesBC,CAandABofthetriangleABC.ProvethatthecircumcirclesofthetrianglesABC,BACandCABhaveacommonpoint.ProvethatthepropertyholdsevenifthepointsA,BandCarecollinear.

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).

LetABCDbeaconvexquadrilateral.ProvethatAB.CD+AD.BC=AC.BDifandonlyifABCDiscyclic(Ptolemystheorem).

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.

IntheinteriorofaquadrilateralABCD,consideravariablepointP.ProvethatifthesumofdistancesfromPtothesidesisconstant,thenABCDisaparallelogram.

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.

LetABCDbeaconvexquadrilateralandletEandFbethepointsofintersectionsofthelinesAB,CDandAD,BC,respectively.ProvethatthemidpointsofthesegmentsAC,BD,andEFarecollinear.

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.

Letd,dbetwononparallellinesintheplaneandletk>0.Findthelocusofpoints,thesumofwhosedistancestodanddisequaltok.

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.

LetABCDbeaconvexquadrilateralandletk>0bearealnumber.FindthelocusofpointsMinitsinteriorsuchthat[MAB]+2[MCD]=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].

LetABCDbeaconvexquadrilateral.FindthelocusofpointsMinitsinteriorsuchthat[MAB]=2[MCD].

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.

LetABCDbeaconvexquadrilateralandMapointinitsinteriorsuchthat[MAB]=[MBC]=[MCD]=[MDA].ProvethatoneofthediagonalsofABCDpassesthroughthemidpointoftheotherdiagonal.

Question Number 16053    Answers: 0   Comments: 1

number of positive integers a and b and c satisfying a^b^c b^c^a c^a^b =5abc

numberofpositiveintegersaandbandcsatisfyingabcbcacab=5abc

Question Number 16014    Answers: 1   Comments: 0

A cirlce is drawn to touch the sides of a triangle whose sides are 12cm,10cm,and 9cm. Find the radius of the circle.

Acirlceisdrawntotouchthesidesofatrianglewhosesidesare12cm,10cm,and9cm.Findtheradiusofthecircle.

Question Number 15987    Answers: 1   Comments: 1

Question Number 15982    Answers: 0   Comments: 3

Question Number 15969    Answers: 0   Comments: 17

Question Number 15919    Answers: 0   Comments: 1

multiply 3x+4y+5x−8y

multiply3x+4y+5x8y

Question Number 15917    Answers: 1   Comments: 1

Question Number 15908    Answers: 1   Comments: 7

Question Number 15904    Answers: 2   Comments: 1

Question Number 15784    Answers: 1   Comments: 1

Question Number 15782    Answers: 1   Comments: 1

Question Number 15700    Answers: 1   Comments: 1

Question Number 15699    Answers: 1   Comments: 0

3+5x=58

3+5x=58

Question Number 15672    Answers: 1   Comments: 0

solve the equation: {5^x } +5x=140 please show workings.....

solvetheequation:{5x}+5x=140pleaseshowworkings.....

Question Number 15642    Answers: 1   Comments: 0

Question Number 15572    Answers: 2   Comments: 1

Question Number 15567    Answers: 2   Comments: 3

Question Number 16613    Answers: 1   Comments: 2

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