[/quote]
,
and 
such that 
.
are positive isn't necessary.
be the orthocenter of an acute-angled triangle 
.
centered at the midpoint of 
and passing through 
and 
.
,
,
and 
.
be a triangle, and let 
be a point on side 
.
at 
and ray 
at 
. The circumcircle of triangle 
intersects the circumcircle 
of triangle 
distinct from point 
.
and 
intersect 
and 
respectively.
to 
.
is extended beyond 
to 
so that 
.
,
be a cyclic quadrilateral. Let 
be the midpoint of the arc 
and 
and the lines 
and 
meet at 
.
and 
.
and 
respectively. Denote by 
the midpoint of the side 
and 
intersect at 
.
and 
have a common point lying on the side 
and 
be concentric circles, with 
)
and 
.
be a trapezoid with parallel sides 
.
and 
lie on the line segments 
and 
,
.
and 
on the line segment 
satisfying ![\[\angle{APB} = \angle{BCD}\qquad\text{and}\qquad \angle{CQD} = \angle{ABC}.\]](http://latex.artofproblemsolving.com/b/1/a/b1ae208955104081bafe221832509d9dd20eeb83.png)
Prove that the points 
and 
are concyclic.
q
answer choices. A participant who answers all questions correctly will receive a gift. To ensure that at least one member of my family answers all questions correctly, how many family members need to take the quiz?
,
and altitude 
.
.
meets 
meets 
be the orthogonal projection of 
meets 
.
.
liters of milk. Then, 
Y by
I have two questions for you to solve.
However, the particular question I am asking here is
Given the 1st question has answer x, find an easy solution to 2nd question in terms of x.
Essentially are these the same question, or is it a coincidence that the answers are similar?
First question, Equilateral triangle ABC has circle O tangent to BC and passing through A. Circle O intersects AB and AC at D and E, with AD=10 and AE=11. What is the length of BC?
Second question, Quadrilateral ABCD has AB=3, AD=4, B and D are right angles, C is 60 degrees.
What is the length of AC?
However, the particular question I am asking here is
Given the 1st question has answer x, find an easy solution to 2nd question in terms of x.
Essentially are these the same question, or is it a coincidence that the answers are similar?
First question, Equilateral triangle ABC has circle O tangent to BC and passing through A. Circle O intersects AB and AC at D and E, with AD=10 and AE=11. What is the length of BC?
Second question, Quadrilateral ABCD has AB=3, AD=4, B and D are right angles, C is 60 degrees.
What is the length of AC?
with center 
and radius 
.
,
,
.
satisfies these conditions. We are given that this circle is tangent with the line 
,
is the side of the triangle and therefore 
We take the discriminant equal to 
to solve for 
