is the incentre and point 
is defined such that 
and 
is the perpendicular bisector of 
.
as 
,
and 
as 
.
.
are there such that 
and 
?
, where 
is a real number. Prove that for all positive 
then prove
and 
Prove that
Let 
Prove that
Let 
Prove that
Let 
Prove that
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 
.
Y by Mango247
Consider a tetrahedron 
whose altitude 
passes through the intersection point of altitudes of triangle 
.
a) Prove that the opposite edges of this tetrahedron are perpendicular.
b) Prove that the lines on which the altitudes of this tetrahedron lie pass through one point.
c) Find out whether the converse is true: “If the lines on which lie the altitudes of the tetrahedron pass through one point, then the bases of the altitudes of this the tetrahedron are the points of intersection of the heights of the corresponding faces. ''
whose altitude 
passes through the intersection point of altitudes of triangle 
.a) Prove that the opposite edges of this tetrahedron are perpendicular.
b) Prove that the lines on which the altitudes of this tetrahedron lie pass through one point.
c) Find out whether the converse is true: “If the lines on which lie the altitudes of the tetrahedron pass through one point, then the bases of the altitudes of this the tetrahedron are the points of intersection of the heights of the corresponding faces. ''
