Consider triangle RST. In this triangle ST>RT and point Q is on the side RS.
Draw the height TH to the side RS. There are two cases where point H can lie.
1 case: Point H lies to the right from points S and R (see first diagram).
Now consider two right triangles STH and QTH.
In triangle STH, by the Pythagorean theorem,
[tex]ST^2=TH^2+SH^2.[/tex]
In triangle QTH, by the Pythagorean theorem,
[tex]QT^2=TH^2+QH^2.[/tex]
Subtract these two equations:
[tex]ST^2-QT^2=SH^2-QH^2.[/tex]
In this case SH>QH, then [tex]SH^2-QH^2>0,[/tex] thus [tex]ST^2-QT^2>0[/tex] and, consequently, [tex]ST^2>QT^2\Rightarrow ST>QT.[/tex]
2 case: Point H lies between points S and R (see second diagram).
Consider two right triangles STH and QTH.
In triangle STH, by the Pythagorean theorem,
[tex]ST^2=TH^2+SH^2.[/tex]
In triangle QTH, by the Pythagorean theorem,
[tex]QT^2=TH^2+QH^2.[/tex]
Subtract these two equations:
[tex]ST^2-QT^2=SH^2-QH^2.[/tex]
In this case, the height falls into point H that divides side SR into two parts SH and RH. Note that if
- triangle is isosceles, then ST=RT and the height TH is also the median. This means that SH=RH;
- ST>RT, then SH>RH
- ST<RT, then SH<RH.
You have that ST>RT, then SH>RH and SH>QH.
Therefore, [tex]SH^2-QH^2>0,[/tex] thus [tex]ST^2-QT^2>0[/tex] and, consequently, [tex]ST^2>QT^2\Rightarrow ST>QT.[/tex]
