by bin_sherlo, May 11, 2025, 7:21 PM 
is a pentagon whose vertices lie on circle 
where 
. Let 
and 
intersect at 
, 
meet 
at 
. 
is the midpoint of arc 
on , not containing 
. If 
holds, then what is the value of 
?by AlperenINAN, May 11, 2025, 7:15 PM 
be a cyclic quadrilateral and let the intersection point of lines and 
be 
. Let the points 
and 
be arbitrary points on sides and respectively, which satisfy the conditions
Prove that 
.by SomeonecoolLovesMaths, May 11, 2025, 11:28 AM ![[asy]
/* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki go to User:Azjps/geogebra */
import graph; size(10cm);
real labelscalefactor = 0.5; /* changes label-to-point distance */
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pen dotstyle = black; /* point style */
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draw((-3.0139955173701907,1.1764654463952184)--(0.7266612107598007,1.3716679271692873), linewidth(2) + linetype("4 4"),EndArrow(6));
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dot((3.32,-2.78),dotstyle);
label("$C$", (3.4,-2.58), NE * labelscalefactor);
dot((-1.4789869126960866,-1.8547454538503692),dotstyle);
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dot((0.726661210759801,1.3716679271692873),dotstyle);
label("$E$", (0.8,1.58), NE * labelscalefactor);
dot((-3.0139955173701907,1.1764654463952184),dotstyle);
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[/asy]](http://latex.artofproblemsolving.com/6/4/a/64a0c503b7bfc56ffb6d38210fd6e0e1e01df376.png)
by SomeonecoolLovesMaths, May 11, 2025, 11:16 AM 
satisfy the equality, 
prove that the triangle must have a right angle.by k.vasilev, Apr 23, 2019, 6:51 PM be an acute-angled triangle with 
A circle passes through 
and 
and crosses the segments and 
again at 
and 
respectively. The circumcircles of 
and meet each other at points 
and 
The segments 
and 
intersect at 
Let 
and 
be the reflections of 
in the lines 
and 
respectively. Prove that the points 
and are concyclic.by parmenides51, Dec 11, 2018, 8:17 PM be an acute triangle with 
. be 
the circumcircle circumscribed to the triangle and 
the midpoint of the smallest arc of this circle. Let and points of the segments and respectively such that 
. Let 
be the second intersection point of the circumcircle circumscribed to 
with . Let and 
be the intersections of lines 
and 
with other than , respectively. Let 
and be the intersection points of lines 
and 
with lines and respectively. Show that the 
line passes through the midpoint of by gobathegreat, Sep 16, 2018, 12:44 PM 
be a center of circle which passes through vertices of quadrilateral , which has perpendicular diagonals. Prove that sum of distances of point to sides of quadrilateral is equal to half of perimeter of .by Sayan, Mar 31, 2013, 6:28 AM ![\[\frac{\sin^2A+\sin^2B+\sin^2C}{\cos^2A+\cos^2B+\cos^2C} = 2\]](http://latex.artofproblemsolving.com/9/2/a/92a524bbdca92e9dd4091275714bf346bad96cd7.png)
by Zhero, Jul 5, 2012, 1:49 AM be a triangle with circumcircle , incenter 
, and -excenter 
. Let the incircle and the -excircle hit at and , respectively, and let be the midpoint of arc without . Consider the circle tangent to at and arc 
at 
. If 
intersects again at , prove that 
and 
meet on .Stay insane,Coz it's your will, labour and pain,which takes you to the top of the mountain.
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