### 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

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

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