be a convex pentagon such that 
.
inside 
and 
.
intersect lines 
and 
at points 
,
occur on their line in that order. Let line 
intersect 
at points 
and 
,
occur on their line in that order. Prove that the points 
lie on a circle.
,
.
such that its decimal representation contains no figures besides these two. Is this possible for each teacher’s choice? (4 marks)
and for all 
,
Determine the smallest 
such that 
.
Y by Adventure10, Mango247
[geogebra]831c935d2f9aa8f009c870e6a3c58466a6ef3cbb[/geogebra]
GIVEN:
A trapezoid
, where 
and 
are right angles.
let
, and 
PROBLEM:
In terms of
and 
, find
(a) the radius
of the inscribed circle
(b) the area of the trapezoid
(c) the perimeter
of the trapezoid
GIVEN:
A trapezoid
, where 
and 
are right angles.let
, and 
PROBLEM:
In terms of
and 
, find(a) the radius
of the inscribed circle(b) the area
of the trapezoid(c) the perimeter
of the trapezoid
be the incenter, 
,
and 
.![\[x^2=(H-r)^2+r^2\\ y^2=(h-r)^2+r^2\\ z^2=(2r)^2+(H-h)^2\\ z^2=x^2+y^2\]](http://latex.artofproblemsolving.com/2/e/6/2e62c216252a5a4c310ab9ef4bb2380e21a777d8.png)
.![\[(H-r)^2+r^2+(h-r)^2+r^2=4r^2+(H-h)^2\]](http://latex.artofproblemsolving.com/f/5/5/f55fffd179f62332420d2d147a7c68e2e8f81b25.png)
![\[r=\frac{Hh}{H+h}.\]](http://latex.artofproblemsolving.com/1/4/6/146ef569129396259855b2438a0475e781f52fa0.png)
![\[A=\frac{2r(H+h)}{2}=Hh.\]](http://latex.artofproblemsolving.com/4/6/4/4649b299ba127f704c85f9b48cfd54ffb84029d5.png)
the perimeter is![\[P=2r+H+h+z=2(H+h).\]](http://latex.artofproblemsolving.com/e/2/6/e2676476385351d26a6343d290f5d12423c3cb63.png)
