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tags: homeschooling, math puzzles, geometry, julius
---

# Geometry without Algebra

(under construction)

## Introduction

A collection of geometry problems. Starting from the beginning, roughly in order of difficulty. Solutions to later problems build on knowledge of earlier ones.

I plan to select only problems that do not involve measurements of lengths and angles and require no calculations with numbers. Some problems are easier with a little bit of algebra (rearranging formulas with lengths and angles) but no problem here requires this.

## Problems

### Basic Properties

The following can all be solved using some basic facts about the existence of parallels, about angles and parallels (Z-angles) and about similar triangles. [^similar]

[^similar]: Two triangles are similar, if they have the same angles. Then one can be obtained from the other by translation, rotation, flipping and enlargement. By flipping I mean that you cut it out from paper and turn it upside down.

- Prove that the sum of the angles in a triangle equals two right angles.

- Prove the theorem of Thales stating that any triangle inscribed in a circle, with one side being the diameter, is has a right angle:


- Find a generalisation of Thales's theorem that allows for the case where none of the sides of the triangle is a diameter.


One may want analyse the green and the red situation separately, but there is also a general statement involving both. It can be formulated as a statement about quadrilaterals inscribed in a circle.

- The picture below shows a square inscribed in a circle.
Deform the square to some quadrilateral by moving the four corners so that they stay on the circle. The green angles are going to change. But certain pairs of angles remain equal to each other. Which ones? Why?

- Draw a triangle with a right angle (and a perpendicular dropped through the right angle)
on paper, twice. Cut out the three triangles. Show that they are similar by aligning them in the right way. Prove that they must be similar.

Use the last result to show that in the picture $a\cdot b = c\cdot c$. ($c$ is also known as the *geometric mean* of $a$ and $b$.)

Use the similarity of the triangles (and a tiny bit of algebra) to prove the Pythagorean theorem.

- Draw any two lines through a circle so that they intersect inside the circle as in the picture. 
Show that $a\cdot b = c\cdot d$. (In other words, $a,b$ and $c,d$ have the same geometric mean.)

- This is similar to the above, but now the intersection is outside of the circle.
Show that $\overline{PA}\cdot\overline{PB}=\overline{PC}\cdot\overline{PC}$. [^PAC]

[^PAC]: There is a more geometric way of expressing the same fact. Show that after flipping the triangle $\it PAC$ (and aligning the angles at $P$) the line $CA$ is parallel to the line $BD$.

- ...

## References

The volume Geometry in the series Art of Problem Solving has many more problems.