*Notes to a video lecture on http://www.unizor.com*

__Geometry+ 05__

*Problem A*

Construct a quadrilateral

*by its 4 sides*

**ABCD***,*

**AB***,*

**BC***,*

**CD***and an angle*

**DA***between opposite sides*

**φ***and*

**AB***.*

**CD***Hint A*

Find point

*such that*

**P***is parallel and congruent to*

**BP***.*

**CD**Consider Δ

*.*

**ABP***Problem B*

Given a circle of radius

*and*

**R***n*-sided regular polygon inscribed into it.

Let

*be any point on this circle.*

**P**Find a sum of squares of distances from this point

*to all vertices of a polygon.*

**P***Hint B*

(a) Geometrical solution for even number

*of vertices of a regular polygon can be obtained by adding pairs of distances from*

**n***to*

**P**

**i**^{th}and (

*)*

**i+**½**n**^{th}vertices.

(b) General solution can be obtained if using vectors from the center of a circle to all its vertices and to point

*.*

**P***Answer*

Sum of squares of distances from point

*to all vertices of a polygon equals to*

**P***.*

**2nR²***Problem C*

Given an equilateral triangle Δ

*.*

**ABC**Extend side

*beyond vertex*

**AC***to point*

**C***and build another equilateral triangle*

**D***with point*

**CDE***on the same side from*

**E***as point*

**AD***.*

**B**Connect points

*and*

**A***. Let point*

**E***be a midpoint of segment*

**M***.*

**AE**Connect points

*and*

**B***. Let point*

**D***be a midpoint of segment*

**N***.*

**BD**Prove that triangle Δ

*is equilateral.*

**CMN***Hint C*

Triangles Δ

*and Δ*

**ACE***are congruent.*

**BCD**
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