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8 votes
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Largest Area Triangle in the Vesica Piscis

EDIT. I'm inserting here a purely geometrical solution, the original reasoning can be seen at the end. I'll repeatedly make use of the following result: if we have a line $r$ and an arc of circle $\...
Intelligenti pauca's user avatar
4 votes

Largest Area Triangle in the Vesica Piscis

For triangles with an edge parallel to the line connecting the centers of the circles, the largest is shown in the image below. I expect this to be the largest in general.
Daniel Mathias's user avatar
3 votes
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Is there is a formula to calculate the coordinates of the orthocenter of a triangle?

For info on tens of thousands of triangle centers, consult Clark Kimberling's Encyclopedia of Triangle Centers; in particular, the orthocenter entry $X(4)$ gives the following barycentric coordinates: ...
Blue's user avatar
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3 votes

Is there is a formula to calculate the coordinates of the orthocenter of a triangle?

There is a formula that is easier to remember, using complex numbers. If the complex coordinates of the points are $z_1$, $z_2$, $z_3$ then we have the center of the circumscribed circle $o$ and the ...
orangeskid's user avatar
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3 votes
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Determine the angle $\angle DEC$ in a triangle (Euclidean Geometry)

Let $F$ be the point symmetric to $C$ with respect to the bisector of angle $CBD$. Then $F$ lies on $BD$ and $$\angle DFC = 90^\circ - \frac 12 \angle CBF = 90^\circ - \alpha = \angle ADB = \angle CDF,...
timon92's user avatar
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2 votes

Largest Area Triangle in the Vesica Piscis

Consider a bounded and closed (i.e. compact) region in the plane, not contained in a line. There exists a triangle with largest area with vertices in the figure, $\Delta ABC$. Now if we keep $B$, $C$ ...
orangeskid's user avatar
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1 vote

Is there is a formula to calculate the coordinates of the orthocenter of a triangle?

In figure O is circumcenter and K is its reflection over BC.Since we have coordinates of vertices we can calculate length of the sides a, b and c and also R the radius of circum circle.We use the fact ...
sirous's user avatar
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1 vote

Is there is a formula to calculate the coordinates of the orthocenter of a triangle?

Yet an other answer... Let $(x,y)$ be the vector of coordinates of the orthocenter $H$ of the triangle $A_1A_2A_3$, where $A_j=(x_j,y_j)$ is (seen as a vector) with given components. Then the ...
dan_fulea's user avatar
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1 vote

Proof using Converse of Thales Theorem for isosceles right-angled triangle

It is not clear what you want, but here is a simple proof: observe that $\angle ADF=\angle DAF=\pi/4$ and therefore $\angle AFD$ is right. So $AF$ is an altitude. Obviously $EC$ is an altitude. ...
GReyes's user avatar
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1 vote
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Knowing a side, the inradius, and the circumradius of a triangle, find the other two sides

$$\tan (\frac x2) = \frac{\sin (x)}{1+\cos(x)} \implies \tan (\frac{\alpha}2) = \frac{8}{17+15} = \frac1{4}$$ $$\tan(\frac{\alpha}2) = \frac r{s-a} \iff s-16 = 4 \cdot 6 \iff s = 40$$ $$a+b+c = 80 \...
hellofriends's user avatar
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