State with reasons that angle ACB is bisected by line OC

[mathlearn]

Here is the Problem

View attachment 6180

Here is the construction, Hoping that I have done it correct "State with reasons that $\angle$ ACB is bisected by line OC"

View attachment 6181

Many THanks (Party)

 

[Greg]

AB is given as 6 cm and the circle should be tangent to AB.

 

[mathlearn]

AB is given as 6 cm and the circle should be tangent to AB.

Yes I updated the diagram correcting that error :) As AB = AC as they are tangents drawn to the circle from external point A. $\therefore \angle{ABC}=\angle{ACB}$

But still how can it be proved that
State with reasons that $\angle {ACB}$ is bisected by line OC"

 

[Greg]

Let $X$ be any point outside of a circle $C$. Let the points $Y$, $Z$ be the points of tangency at the circle $C$ and the lines $XY$ and $XZ$. What can be said about triangles $XYC$ and $XZC$ ?

 

[mathlearn]

Here is the Problem
Here is the construction, Hoping that I have done it correct "State with reasons that $\angle$ ACB is bisected by line OC"
Many THanks (Party)

Let $X$ be any point outside of a circle $C$. Let the points $Y$, $Z$ be the points of tangency at the circle $C$ and the lines $XY$ and $XZ$. What can be said about triangles $XYC$ and $XZC$ ?

Thanks but I doubt whether the Line $OC$ bisects angle $ACB$ which is really inside the triangle ? (Thinking) Have I done the construction wrong? Is the type of the circle correct?

 

[Greg]

See post #2.

The relation that is to be found is true in general, which is what I was trying to get at in post #4. Make the necessary corrections and apply the hints I gave to state your answer (with reasons).

 

[mathlearn]

See post #2.

The relation that is to be found is true in general, which is what I was trying to get at in post #4. Make the necessary corrections and apply the hints I gave to state your answer (with reasons).

View attachment 6185

Many THanks :)

 

[Greg]

View attachment 6186

Like that! Does that help?

 

[mathlearn]

:D Thank you very much , I was struggling a long here for around a day or two

The sides was produced in the other direction

And Greg I think the reason for the bisection is the congruence of those two triangles due to the radius of the circle at the tangent and the common line CD and the 90 degree angle formed at the tangent line and the radius

Many THanks (Smile)

 

[Greg]

That's correct. Good work! :)