Geometry Set

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  1. Let \(P\) and \(Q\) be points on the side \(AB\) of the triangle \(ABC\) (with \(P\) between \(A\) and \(Q\)) such that \(\angle{ACP}=\angle{PCQ}=\angle{QCB}\), and let \(AD\) be the angle bisector of \(\angle{BAC}\) with \(D\) on \(BC\). Line \(AD\) meets lines \(CP\) and \(CQ\) at \(M\) and \(N\) respectively. Given that \(PN=CD\) and \(3\angle{BAC}=2\angle{BCA}\), prove that triangles \(CQD\) and \(QNB\) have the same area. (Belarus 1999)

  2. Let ABCABC be a triangle and let ω\omega be its incircle. Let D1D_1 and E1E_1 be the tangency points of ω\omega with sides BCBC and ACAC, respectively. Let D2D_2 and E2E_2 be points on sides BCBC and ACAC, respectively, such that CD2=BD1CD_2=BD_1 and CE2=AE1CE_2=AE_1, and let PP be the intersection point of segments AD2AD_2 and BE2BE_2. Circle ω\omega intersects segment AD2AD_2 in two points, being the closest one to AA denoted by QQ. Prove that AQ=D2PAQ=D_2P.

  3. Points A,B,C,DA, B, C, D are four consecutive vertices of a regular polygon such that 1AB=1AC+1AD\dfrac{1}{AB}=\dfrac{1}{AC}+\dfrac{1}{AD}. How many sides does the polygon have?

#Geometry #Polygons #Triangles #Areas

Note by José Marín Guzmán
6 years, 6 months ago

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Comments

3)The question number 3 is easy to solve with complex numbers.But also found a geometric solution.

Let the polygon be nn- sided.

Suppose the vertex after DD be EE.Hence the 55 consecutive vertices are A,B,C,D,EA,B,C,D,E ( arranged anticlockwise from A to E,say)

Firstly,1AB=1AC+1AD\frac{1}{AB}=\frac{1}{AC}+\frac{1}{AD} gives

AC.AD=AB.AD+AB.ACAC.AD=AB.AD+AB.AC...(1)(1)

One can easily show that quadrilateral ACDEACDE is cyclic.

So,by Ptolemy's theorem, AE.CD+AC.ED=AD.ECAE.CD+AC.ED=AD.EC...(2)(2)

Also,since the polygon is regular,we have AB=CD=EDAB=CD=ED and EC=ACEC=AC.

So,from (1)(1), AB.AD+AB.AC=AC.AD=CE.AD=AE.CD+AC.ED=AE.AB+AC.ABAB.AD+AB.AC=AC.AD=CE.AD=AE.CD+AC.ED=AE.AB+AC.AB

Hence,AD=AEAD=AE.

Since,these two diagonals are equal,number of vertices between AA and EE(moving clockwise) equals the number of vertices between AA and DD(moving anti-clockwise).

Hence,n−5=2n-5=2 or n=7n=7.

Souryajit Roy - 6 years, 6 months ago
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