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# Puzzles

## Archive

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## Is it equilateral?

In the diagram below, $$ABDC$$ is a square. Angles $$ACE$$ and $$BDE$$ are both 75°.
Is triangle $$ABE$$ equilateral? Why/why not?

## 20 December

Earlier this year, I wrote a blog post about different ways to prove Pythagoras' theorem. Today's puzzle uses Pythagoras' theorem.
Start with a line of length 2. Draw a line of length 17 perpendicular to it. Connect the ends to make a right-angled triangle. The length of the hypotenuse of this triangle will be a non-integer.
Draw a line of length 17 perpendicular to the hypotenuse and make another right-angled triangle. Again the new hypotenuse will have a non-integer length. Repeat this until you get a hypotenuse of integer length. What is the length of this hypotenuse?

## Cutting corners

The diagram below shows a triangle $$ABC$$. The line $$CE$$ is perpendicular to $$AB$$ and the line $$AD$$ is perpedicular to $$BC$$.
The side $$AC$$ is 6.5cm long and the lines $$CE$$ and $$AD$$ are 5.6cm and 6.0cm respectively.
How long are the other two sides of the triangle?

## Two triangles

Source: Maths Jam
The three sides of this triangle have been split into three equal parts and three lines have been added.
What is the area of the smaller blue triangle as a fraction of the area of the original large triangle?

## Equal side and angle

In the diagram shown, the lengths $$AD = CD$$ and the angles $$ABD=CBD$$.
Prove that the lengths $$AB=BC$$.

## Arctan

Prove that $$\arctan(1)+\arctan(2)+\arctan(3)=\pi$$.

## Square deal

This unit square is divided into four regions by a diagonal and a line that connects a vertex to the midpoint of an opposite side. What are the areas of the four regions?

## Half an equilateral triangle

Source: Maths Jam
What is the shortest straight line which bisects an equilateral triangle?