AoPS Academy
with circumcenter 
and orthocenter 
, let 
be an arbitrary point on the circumcircle of triangle such that does not lie on line 
and that line 
is not parallel to line 
. Let 
be the point on the circumcircle of triangle such that 
is perpendicular to , and let 
be the point on line 
such that 
. Let 
and 
be the points on the circumcircle of triangle such that 
is a diameter, and 
and 
are parallel. Let 
be the midpoint of 
.
and 
are perpendicular.
and 
are perpendicular.
with 
, a tangent to the circumcircle of at point 
intersects the extension of 
at point 
. 
is the midpoint of 
, and 
intersects the circumcircle of at 
. Prove that 
.
be an isosceles right triangle with 
. Point is the midpoint of the side ![$[AC]$](http://latex.artofproblemsolving.com/0/9/3/0936990e6625d65357ca51006c08c9fe3e04ba0c.png)
, and point ![$E \in [AC]$](http://latex.artofproblemsolving.com/c/c/b/ccb6bb6e6099e012693f028e6a34b7319643ee6c.png)
is so that 
. Calculate 
.
is a symmedian of 
, then please post it, because my solution uses Cartesian coordinates lol. be tangent intersection through 
and 
to circumcircle of .![\[\angle AEB + \angle ADB=2\angle ACB + \angle EBC+ \angle DBC=90^{\circ}+ \angle EBC+ \angle DBC\]](http://latex.artofproblemsolving.com/0/0/5/00583c32ced0d40dc49d1ce2a90b9d8317990bcc.png)
We will show that is a symmedian of . That can be shown with Cartesian coordinates.
and 
, which implies that 
, which implies that 
. By similar triangles we see that line slope is 
, thus line is described in Cartesian plane as 
. Plugging in point 's 
-coordinate, we get point 's 
-coordinate, thus 
, and are collinear, which implies that is symmedian of (by the Symmedian Lemma). is a symmedian of , thus 
(by the symmedian property), thus![\[\angle AEB + \angle ADB=90^{\circ}+ \angle EBC+ \angle DBC=90^{\circ}+ \angle EBC+ \angle ABE=90^{\circ}+ \angle ABC=135^{\circ}\]](http://latex.artofproblemsolving.com/1/4/b/14bcddcbcb5f85fd1bc3622c3b5dfa22ce4269c3.png)
Answer. 
![[asy]
import graph;
real labelscalefactor = 0.5; /* changes label-to-point distance */
pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */
pen dotstyle = black; /* point style */
real xmin = -0.6649446192944306, xmax = 13.383313846642052, ymin = -11.929015277274319, ymax = 14.073351324272796; /* image dimensions */
pen qqwuqq = rgb(0,0.39215686274509803,0); pen cqcqcq = rgb(0.7529411764705882,0.7529411764705882,0.7529411764705882);
draw((5.672789010886572,5.672789010886572)--(6,5.345578021773145)--(6.327210989113428,5.672789010886572)--(6,6)--cycle, linewidth(2) + qqwuqq);
/* draw grid of horizontal/vertical lines */
pen gridstyle = linewidth(0.7) + cqcqcq; real gridx = 2, gridy = 2; /* grid intervals */
for(real i = ceil(xmin/gridx)*gridx; i <= floor(xmax/gridx)*gridx; i += gridx)
draw((i,ymin)--(i,ymax), gridstyle);
for(real i = ceil(ymin/gridy)*gridy; i <= floor(ymax/gridy)*gridy; i += gridy)
draw((xmin,i)--(xmax,i), gridstyle);
/* end grid */
Label laxis; laxis.p = fontsize(10);
xaxis(xmin, xmax, Ticks(laxis, Step = 2, Size = 2, NoZero),EndArrow(6), above = true);
yaxis(ymin, ymax, Ticks(laxis, Step = 2, Size = 2, NoZero),EndArrow(6), above = true); /* draws axes; NoZero hides '0' label */
/* draw figures */
draw((0,0)--(6,6), linewidth(1));
draw((6,6)--(12,0), linewidth(1));
draw((12,0)--(0,0), linewidth(1));
draw((0,0)--(9,3), linewidth(1));
draw(circle((6,0), 6), linewidth(1));
draw((xmin, 0*xmin + 6)--(xmax, 0*xmax + 6), linewidth(1)); /* line */
draw((12,ymin)--(12,ymax), linewidth(1)); /* line */
draw((xmin, 0.5*xmin + 0)--(xmax, 0.5*xmax + 0), linewidth(1)); /* line */
/* dots and labels */
dot((0,0),dotstyle);
label("$B$", (0.3385024139867468,-0.5420728561270076), NE * labelscalefactor);
dot((6,6),dotstyle);
label("$A$", (5.792018899210538,6.4602423109003615), NE * labelscalefactor);
dot((12,0),dotstyle);
label("$C$", (12.379866813360875,0.04690692427716356), NE * labelscalefactor);
dot((9,3),linewidth(4pt) + dotstyle);
label("$D$", (9.282269449753763,3.384459013234134), NE * labelscalefactor);
dot((8,4),dotstyle);
label("$E$", (7.798912965772892,4.627860771865162), NE * labelscalefactor);
dot((12,6),linewidth(4pt) + dotstyle);
label("$P$", (12.09628395612924,6.176659453668723), NE * labelscalefactor);
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);
[/asy]](http://latex.artofproblemsolving.com/5/1/f/51f2a8178284e54e5136c3ad3f985d0b484e6b58.png)
prove that![$\frac{a}{b} + \sqrt {\frac{b}{c}} + \sqrt[3]{{\frac{c}{a}}} > \frac{5}{2}$](http://latex.artofproblemsolving.com/6/a/6/6a657cb647e279b0745e3189c3f9d54aa25bf976.png)
over 
.
then 
,
.
,
.