# What is the way to write geometry proofs?

Dec 6, 2015

A recommendation is below.

#### Explanation:

For instance,
"Given an isosceles triangle".

Then specify, what is to be proven.
For instance,
"Prove that the median to its base is also an altitude".

Then it is useful to draw a picture of what's given with labels to each element involved.
For instance,

Add elements needed for the proof you'd like to offer.
For instance,
"Draw a median $B M$ from vertex $B$ to base $A C$, so $A M = M C$".

Then specify logical statements, each having a known fact, derived conclusion and the logical basis this conclusion is founded upon in some (not necessarily this) order.
It is important to preserve the logic of derivation that leads, step by step, from given information to a statement that necessary to prove, so each subsequent step is based only on known and proven statements.
Of course, certain fundamental theorems studied before should be considered as proven and can be a foundation for logical derivation.
For instance,

(1) Triangles $\Delta A B M$ and $\Delta M B C$ are congruent because all three sides of one correspondingly congruent to sides of another:
$A B = B C$ because $\Delta A B C$ is isosceles;
$A M = M C$ because $B M$ is a median and, therefore, divides side $A C$ in halves;
side $B M$ is common for these triangles.

(2) Since triangles $\Delta A B M$ and $\Delta M B C$ are congruent and angles $\angle A M B$ and $\angle B M C$ are opposite to congruent sides $A B$ and $B C$, these angles are congruent.

(3) Since angles $\angle A M B$ and $\angle B M C$ * are congruent and represent a linear pair with their sum being equal to ${180}^{o}$, each must be equal to ${180}^{o} / 2 = {90}^{o}$, that is $B M \bot A C$, that is $B M$ is an altitude.

End of proof