#### What is Euclid’s Theorem for triangles? The proof of Euclid’s theorem and examples.

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### Euclid’s Theorem;

We have similar triangles:

- ΔABC ∼ ΔDBA ∼ ΔDAC

and we have also found two of Euclid’s theorems using similar triangles.

- c²=ap and b²=ak

We call these statements Euclid’s right side theorems.

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###### Here are Euclid’s other theorems on right angled triangles :

- ΔDBA ∼ ΔDAC

We call this statement Euclid’s altitude theorem.

- We also have
- Now
- so
- A more simple way of finding this last formula is to look at the area of ΔABC in two different ways.
- If we take [BC] as the base then [AD] is the altitude.

So the area of ΔABC is :

- If we take [AC] as the base then [AB] is the altitude. So the area of ΔABC is :
- Here are the formulas for Euclid’s theorems put together:
- Notice that if we write these formulas in the forms :

then :

- c is the mean proportional between a and p
- b is the mean proportional between a and k
- h is the mean proportional between p and k
- h is the fourth proportional to a, b and c