# Geometry Set

The problems are in no particular order. Please reshare the note if you like what you see, feel free to share comments or ideas and enjoy.

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 $ABC$ be a triangle and let $\omega$ be its incircle. Let $D_1$ and $E_1$ be the tangency points of $\omega$ with sides $BC$ and $AC$, respectively. Let $D_2$ and $E_2$ be points on sides $BC$ and $AC$, respectively, such that $CD_2=BD_1$ and $CE_2=AE_1$, and let $P$ be the intersection point of segments $AD_2$ and $BE_2$. Circle $\omega$ intersects segment $AD_2$ in two points, being the closest one to $A$ denoted by $Q$. Prove that $AQ=D_2P$.

3. Points $A, B, C, D$ are four consecutive vertices of a regular polygon such that $\dfrac{1}{AB}=\dfrac{1}{AC}+\dfrac{1}{AD}$. How many sides does the polygon have? Note by José Marín Guzmán
6 years, 4 months ago

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## Comments

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

Let the polygon be $n$- sided.

Suppose the vertex after $D$ be $E$.Hence the $5$ consecutive vertices are $A,B,C,D,E$ ( arranged anticlockwise from A to E,say)

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

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

One can easily show that quadrilateral $ACDE$ is cyclic.

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

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

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

Hence,$AD=AE$.

Since,these two diagonals are equal,number of vertices between $A$ and $E$(moving clockwise) equals the number of vertices between $A$ and $D$(moving anti-clockwise).

Hence,$n-5=2$ or $n=7$.

- 6 years, 4 months ago

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