# Let $ABCD$ be a cyclic quadrilateral and let $AB$ and $CD$ meet at $E$. Let $M= (EBC)cap (EAD)$. Prove that $OMperp EM$

Mathematics Asked by Raheel on December 16, 2020

Let $$ABCD$$ be a cyclic quadrilateral and let $$AB$$ and $$CD$$ meet at $$E$$. Let $$M= (EBC)cap (EAD)$$. Prove that $$OMperp EM$$

I took midpoint of $$AB$$ as $$M_1$$ and midpoint of $$DC$$ as $$M_2$$ . I noticed that $$(EM_1OM_2)$$ is cyclic and it is enough to show that $$EM_1OM_2M$$ is cyclic. PS: Diagram by @Shubhangi

You are very close! Just note that M is the spiral center of the spiral similarity $$S$$ sending $$AB$$ to $$DC$$ . And hence the spiral similarity $$S$$ also take the midpoint of $$AB$$ to midpoint of $$DC$$.

So $$S:M_1 rightarrow M_2$$

So $$S:BM_1 rightarrow CM_2$$ .

So $$M$$ is the spiral center of the spiral symmetry which takes $$BM_1$$ to $$CM_2$$.

But notice that $$BM_1cap CM_2=E implies M =(EBC) cap (EM_1M_2)$$

So $$M in (EM_1M_2)$$ and by your observation, we get $$Min (EM_1OM_2)$$ , and hence we have $$OMperp EM$$.

Here M is called the miquel point and if we define $$F=BCcap DA$$ , then we have $$Min EF$$ if $$ABCD$$ is cyclic .

Correct answer by Sunaina Pati on December 16, 2020

## Related Questions

### How to check if two functions only touch in one point?

3  Asked on December 13, 2021

### $ker(T)^{bot} = overline{im(T^*)}$ if $T$ is a linear operator between Hilbert spaces

1  Asked on December 13, 2021 by rino

### Partial Fraction Decomposition of $frac{1}{x^2(x^2+25)}$

2  Asked on December 13, 2021 by mjoseph

### How to project (draw) a rectangle on an incline plane

1  Asked on December 13, 2021 by alexander-cska

### How to find a basis of complementary subspace of a subspace not in $mathbb R^n$?

1  Asked on December 13, 2021 by jon-g

### Equality of ideals in $mathbb{Z}[sqrt{-7}]$

1  Asked on December 13, 2021

### An orthonormal basis in a subspace.

1  Asked on December 13, 2021 by ryuta-osawa

### Is there a simple-ish function for modeling seasonal changes to day/night duration and height of the sun?

3  Asked on December 13, 2021 by saganritual

### Parallel transport on the Stiefel manifold

1  Asked on December 10, 2021 by gcc

### Remainder when $^{40}C_{12}$ is divided by $7$.

2  Asked on December 10, 2021 by yash-bhatia

### Showing that for some group it is abelian iff $x • (y • x ^{−1} ) = y$

2  Asked on December 10, 2021 by cheese12345

### Summation involving Gamma function

1  Asked on December 10, 2021 by paranoid

### Probability one random variable is less than another random variable but higher than the same other random variable with a factor

1  Asked on December 10, 2021 by sottoj

### Find x in a multiple arithmetic sequence

1  Asked on December 10, 2021 by billy-senders

### Proof that this is a Primitive Recursive function

2  Asked on December 10, 2021 by user6767509

### Convergence and absolute convergence of an infinite product of terms in $(0,1]$.

1  Asked on December 10, 2021 by lce

### Describing the set ${nin Bbb Nlvert (n>1)wedge (forall x,yin Bbb N)[(xy=n)implies (x=1lor y=1)]}$

2  Asked on December 10, 2021

### What is the permutation of choosing the just 3 balls in a pool of 16 balls?

1  Asked on December 10, 2021 by mahesh87

### Determine number of divisors of $x^n -1$ of degree $k$. Why are they such a number?

1  Asked on December 10, 2021

### why does a set of parametric equations need to be smooth and not cross itself in order to evaluate its arc length?

0  Asked on December 10, 2021 by lightbulb