, where 
is a real number. Prove that for all positive 
then prove
and 
Prove that
Let 
Prove that
Let 
Prove that
Let 
Prove that
is the incentre and point 
is defined such that 
and 
is the perpendicular bisector of 
and 
be the circles passing through 
and 
respectively such that 
is tangent to 
to be a point such that it lies on both the circles 
and 
are perpendicular.
weights, each weighing 
and another 
weights with 
each. Write a algorithm that determines the ways in which a scale can be balanced with a weight 
on the left pan, and display the number of possible solutions. (The weights can be placed on both pans and the program starts with the numbers 
.
with orthocenter 
.
to meet 
.
meets the midpoint of 
.
,
.
with side length of 
.
such that 
.
as 
.
.
as 
respectively, compute the area of 
given the area of 
.
Y by
The two circles have no common points and are outside each other. 
and 
are their external tangents, and 
is internal tangent (points 
and 
lie on the segments and respectively). Prove that 
.
(Sangaku, Japanese temple geometry).
and 
are their external tangents, and 
is internal tangent (points 
and 
lie on the segments and respectively). Prove that 
.(Sangaku, Japanese temple geometry).
This post has been edited 2 times. Last edited by parmenides51, Jul 4, 2021, 4:56 PM
be the tangency points of the circle 
be the tangency points of the circle 
,
,
and 
.
,
.
and 
which means 
.
and 
so 
.
and 
so adding we have 
so 
and hence 
so 
as desired.
