When the last problem in the math competition was too hard for everyone to solve.

$(O)$ is a circle with diameters $B_1C_1$ and $B_2C_2$ and tangents $B_1D$ and $B_2D$ . $A$ is on the larger arc of $BC$ . $AD \cap (O) = \{E\}$ . $DO$ cuts $C_1A$ , $C_1E$ , $C_2A$ , $C_2E$ respectively at $M_1$ , $M_2$ , $N_1$ , $N_2$ . Prove that $M_1N_1 = M_2N_2$ .

#Geometry

Easy Math Editor

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Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

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Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$