Suppose A 1 A 2 A 3 … A n is an n -sided regular polygon such that 1 A 1 A 2 = 1 A 1 A 3 + 1 A 1 A 4 . Determine n , the number of sides of the polygon.

From the given relation, we have A 1 A 2 · A 1 A 3 + A 1 A 2 · A 1 A 4 = A 1 A 3 · A 1 A 4 . . . 1 Also in the cyclic quadrilateral A 1 A 3 A 4 A 5 , by Ptolemy's theorem, we get A 4 A 5 · A 1 A 3 + A 3 A 4 · A 1 A 5 = A 3 A 5 · A 1 A 4 . . . 2 Since A 1 A 2 A 3 … A n is a regular n -sided polygon. So, A 1 A 2 = A 4 A 5 , A 1 A 2 = A 3 A 4 , A 1 A 3 = A 3 A 5 . Thus, equation 2 reduces to A 1 A 2 · A 1 A 3 + A 1 A 2 · A 1 A 5 = A 1 A 3 · A 1 A 4 . . . 3 . Comparing equations 1 & 3 , we get A 1 A 4 = A 1 A 5 Since the two diagonals A 1 A 4 and A 1 A 5 are equal. It follows that there must be the same number of vertices between A 1 and A 4 as between A 1 and A 5 . Since, there are two vertices A 2 & A 3 between A 1 and A 4 . So, there must be two vertices between A 1 and A 5 . Thus, the polygon must be 7 -sided, that is n = 7 . Aliter: If O is the centre of the circle in which A 1 A 2 … A n is inscribed and θ is the angle which each side of the polygon subtends at O , then using the relation 1 A 1 A 2 = 1 A 1 A 3 + 1 A 1 A 4 , obtain an equation in θ . Solve the equation to get θ = 2 π 7 . This means n = 7 .

E M B I B E

Problem Primer for the Olympiad>Chapter 3 - Geometry>Geometry>Q 64

HARD

Olympiad

IMPORTANT

Earn 100

A triangle $A B C$ has incentre $I$ . Its incircle touches the side $B C$ at $T$ . The line through $T$ parallel to $I A$ meets the incircle at $S$ and the tangent to the incircle at $S$ meets sides $A B, A C$ in points $C^{'}, B^{'}$ respectively. Prove that triangle $A B^{'} C^{'}$ is similar to triangle $A B C$ .

Important Questions on Geometry

MEDIUM

Olympiad

IMPORTANT

Problem Primer for the Olympiad>Chapter 3 - Geometry>Geometry>Q 65

Suppose $A_{1} A_{2} A_{3} \dots A_{n}$ is an $n$ -sided regular polygon such that $\frac{1}{A_{1} A_{2}} = \frac{1}{A_{1} A_{3}} + \frac{1}{A_{1} A_{4}}$ . Determine $n$ , the number of sides of the polygon.

HARD

Olympiad

IMPORTANT

Problem Primer for the Olympiad>Chapter 3 - Geometry>Geometry>Q 66

Suppose $A B C D$ is a quadrilateral such that a semicircle with its centre at the midpoint of $A B$ and bounding diameter lying on $A B$ touches the other three sides $B C, C D$ and $D A$ . Show that $A B^{2} = 4 B C \cdot A D$ .

HARD

Olympiad

IMPORTANT

Problem Primer for the Olympiad>Chapter 3 - Geometry>Geometry>Q 67

Let

A B C

be an acute-angled triangle. For any point

P

lying within this triangle, let

D, E, F

denote the feet of the perpendiculars from

P

onto the sides of

B C, C A, A B

respectively. Determine the set of all possible positions of the point

P

for which the triangle

D E F

is isosceles. For which positions of

P

will the triangle

D E F

become equilateral.

HARD

Olympiad

IMPORTANT

Problem Primer for the Olympiad>Chapter 3 - Geometry>Geometry>Q 68

Three congruent circles have a common point

O

and lie inside a triangle such that each circle touches a pair of sides of the triangle. Prove that the incentre and the circumcentre of the triangle and the point

O

are collinear.

HARD

Olympiad

IMPORTANT

Problem Primer for the Olympiad>Chapter 3 - Geometry>Geometry>Q 69

Let

A B C

be a triangle with

∠ A = 90 °

, and

S

be its circumcircle. Let

S_{1}

be the circle touching the rays

A B, A C

and the circle

S

internally. Further let

S_{2}

be the circle touching the rays

A B, A C

and the circle

S

externally. If

r_{1}, r_{2}

be the radii of the circles

S_{1}

and

S_{2}

respectively, show that

r_{1} r_{2} = 4 [A B C]

. (Here,

[]

denotes area of triangle)

MEDIUM

Olympiad

IMPORTANT

Problem Primer for the Olympiad>Chapter 3 - Geometry>Geometry>Q 70

The diagonals $A C$ and $B D$ of a cyclic quadrilateral $A B C D$ meet at right angles in $E$ . Prove that $E A^{2} + E B^{2} + E C^{2} + E D^{2} = 4 R^{2},$ where $R$ is the radius of the circumscribing circle.