Geometrical Proofs

Introduction


Intercept theorem
There are 2 ways to apply the intercept theorem

In the first scenario, you will be able to identify a pair of parallel lines from the question
Upon identifying the pair of parallel lines,
we conclude that the ratio of AD:DB is the same as the ratio of AE:EC

In the second scenario, you identify that that the ratio of AD:DB is the same as the ratio of AE:EC
Subsequently you conclude that the lines DE and BC are parallel


Midpoint Theorem
The Midpoint theorem is a specific case of the Intercept theorem when the ratio of the lengths is 1:1

There are 2 ways to apply the Midpoint Theorem

The first way is to use Midpoint Theorem to show that 2 lines are parallel
From the information given in the question, we will be able to identify that D is the midpoint of AB and E is the midpoint of AC
Given that D and E are midpoints, we will be able to conclude that DE is parallel to BC

The second way is to use Midpoint Theorem to show that E is a midpoint
From the information given in the question, we know that DE is parallel to BC and D is a midpoint
Using the Midpoint theorem, we can conclude that E is the midpoint of AC

Question

Given that:
O is centre of circle, AB is diameter, C lies on circle
AD is tangent to circle at A, EC is tangent to circle at C

i) Prove that triangles AEO and CEO are congruent
Strategy: make use of information given in the question
AD is tangent to circle --> angle OAE is 90 degrees
EC is tangent to circle --> angle OCE is 90 degrees
O is centre of circle --> OA and OC are radii of the circle, hence their lengths are equal
From diagram OE is the common length
Using the RHS property, we can show that triangles AEO and CEO are congruent