1. Proofs in Plane Geometry

Please show me how to prove all the sub-parts except part(a).Thanks.

2. Prove: $\displaystyle (KF)^2=KN \cdot KC$

This follows easily from $\displaystyle \triangle NKF \sim \triangle FKC$

and setting up your corresponding proportional sides:

$\displaystyle \frac{NK}{FK}=\frac{KF}{KC}$

$\displaystyle (KF)^2=NK \cdot KC$

Prove: $\displaystyle KF=KE$

$\displaystyle (KE)^2=KN \cdot KC$ because a $\displaystyle \overline {KE}$ is a tangent and $\displaystyle \overline {KC}$ is a secant intersecting the circle at N. Justification: Secant-Tangent Theorem.

Using the previous conclusion and this, we use substitution to obtain:

$\displaystyle (KF)^2=(KE)^2$

$\displaystyle KF=KE$

3. Originally Posted by masters
Prove: $\displaystyle (KF)^2=KN \cdot KC$

This follows easily from $\displaystyle \triangle NKF \sim \triangle FKC$

and setting up your corresponding proportional sides:

$\displaystyle \frac{NK}{FK}=\frac{KF}{KC}$

$\displaystyle (KF)^2=NK \cdot KC$

Prove: $\displaystyle KF=KE$

$\displaystyle (KE)^2=KN \cdot KC$ because a $\displaystyle \overline {KE}$ is a tangent and $\displaystyle \overline {KC}$ is a secant intersecting the circle at N. Justification: Secant-Tangent Theorem.

Using the previous conclusion and this, we use substitution to obtain:

$\displaystyle (KF)^2=(KE)^2$

$\displaystyle KF=KE$
Last Proof: $\displaystyle (KE)^2=\frac{1}{4}FN \cdot FA$

$\displaystyle (FE)^2=FN \cdot FA$ by Secant Tangent Theorem

$\displaystyle KF=KE$ Previously proved

$\displaystyle FE=2(KE)$

Substituting, $\displaystyle (2KE)^2=FN \cdot FA$

$\displaystyle 4(KE)^2=FN \cdot FA$

Q.E.D. $\displaystyle (KE)^2=\frac{1}{4}FN \cdot FA$