I am trying to solve 2 questions
Is there a way to find an equation for the length of a chord between two tangents in terms of the radius and distance of that chord to the external point?
Is there a way to find an equation for the distance between that chord and the center of the circle in the same terms (radius and distance of chord to external point)?
Please show working/proof
Here is a graphical representation
Lets define some terms:
$w = AB$
$t = OI$
$h = IP$ ie height of $ΔABP$
$r = OB$ ie radius
$AP$ and $BP$ are tangents to the circle
I am wanting to find:
An equation for $w$ in terms of $h$ and $r$ ie "$w = ...$"
An equation for $t$ in terms of $h$ and $r$ ie "$t = ...$"
Is this actually possible??
I have had a go and concluded the following:
$ΔBOP|||ΔAOP$ (radius, shared side + 90deg) (congruent)
$ΔBOP☰ΔIBP$ (shared angle, 3 angles) (similar)
$ΔIBP☰ΔIOB$ ($∠IOB=∠IBP$ [Angle of chord to tangent is half angle of sector created by chord], 3 angles) (similar)
Hence $ΔBOP☰ΔIBP☰ΔIOB$
But I don't no where to go from there
Thanks
UPDATE So using the similar triangles $\Delta IOB \sim \Delta IBP$ we can state: $$ \frac{w}{t}=\frac{h}{w}, \therefore w^2 = th $$
We can also use Pythagoras Theorem on $\Delta IOB$ $$ \frac{w}{t}=\frac{h}{w}, \therefore w^2 = th $$
The hint.
Since $BL$ is an altitude of $\Delta OBP$ and $\measuredangle OBP=90^{\circ},$ we obtain $$w^2=th$$ and $$w^2=h\sqrt{r^2-w^2}.$$ Now, solve the last equation.