# Thread: Proofs in Plane Geometry

1. ## Proofs in Plane Geometry

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

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

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

and setting up your corresponding proportional sides:

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

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

Prove: $KF=KE$

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

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

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

$KF=KE$

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

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

and setting up your corresponding proportional sides:

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

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

Prove: $KF=KE$

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

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

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

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

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

$KF=KE$ Previously proved

$FE=2(KE)$

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

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

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