What is the way to write geometry proofs?

1 Answer
Dec 6, 2015

A recommendation is below.

Explanation:

Start with known (given) information.
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,
Isosceles triangleIsosceles triangle

Add elements needed for the proof you'd like to offer.
For instance,
"Draw a median BMBM from vertex BB to base ACAC, so AM=MCAM=MC".

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 ABM and Delta MBC are congruent because all three sides of one correspondingly congruent to sides of another:
AB=BC because Delta ABC is isosceles;
AM=MC because BM is a median and, therefore, divides side AC in halves;
side BM is common for these triangles.

(2) Since triangles Delta ABM and Delta MBC are congruent and angles /_AMB and /_BMC are opposite to congruent sides AB and BC, these angles are congruent.

(3) Since angles /_AMB and /_BMC * 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 BM_|_AC, that is BM is an altitude.

End of proof