Let's take a concave mirror with $CP$ as its principal axis. Let's assume that a parallel ray $p$ to $CP$ reflects at $A$ to pass through the focal point $F$. Then, $\angle CAF=\angle ACF$, which suggests that $CF=AF$. Let's take another parallel ray $q$ to $CP$ reflects at $B$ to pass through $F$, then $\angle CBF=\angle BCF$, which suggests that $CF=BF$. Hence we can conclude that $AF=BF$ which suggests that $F$ is center of curvature. Then, if $F=C$, $\angle CAP=0$, which is incorrect. So this suggests that all parallel rays reflecting on a concave mirror do not intersect at one point. Then what about the reflection theory?
closed as unclear what you're asking by John Rennie, Kyle Kanos, ACuriousMind♦, yuggib, user10851 Jul 1 '15 at 20:50
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Parallel rays reflecting on a concave mirror do intersect at one point, the focus, if the mirror is a parabola (in 2d plane geometry) or paraboloid (in 3d space geometry).