[Some] Geometry Prerequisites

[Mao]

Quadrilaterals

To show ABCD is a parallelogram, it is sufficient to show that:




To show ABCD is a rectangle, it is sufficient to show that it is a parallelogram, and




To show ABCD is a square, it is sufficient to show that it is a rectangle, and




To show ABCD is a rhombus, it is sufficient to show that it is a parallelogram, and




To show ABCD is a kite, it is sufficient to show that there are two pairs of sides with equal magnitudes but are not parallel.


To show ABCD is a trapezium, it is sufficient to show that there is one pair of parallel sides.






Triangles

A circumscribed triangle is right-angled if one of its side is a diameter.


The others are either dealing with circles or lines, which tend to be quite basic (or not applicable). They're all in my bound reference (somewhere in this board if you look hard enough) if you want them.

[sachinmachin]

thanks mao :]
ive been meaning to go over this..
much appreciated.

[bem9]

for a rhombus, how does the 'diagonals intersect at right angles' property help.
Can this be the only property to show or do you have to show another one as well?

[Mao]

for a rhombus, how does the 'diagonals intersect at right angles' property help.
Can this be the only property to show or do you have to show another one as well?

That property can help. But diagonals can intersect at right angles for any quadrilateral, so it's not usually used in a proof.

For a rhombus, you have to show diagonals intersect at right angles, then all four sides are the same magnitude.

For a kite, you have to show diagonals intersect at right angles, then two pairs are the same magnitude.

[/0]

Some others:

Finding circles going through points
The perpendicular bisectors of the sides of a triangle intersect at its circumcentre.
i.e. To find the centre of the circle going through 3 points, construct a triangle from these points and find the intersection of its perpendicular bisectors.

If 3 points are symmetrical about an axis, then the circumcentre will lie on this axis.
e.g. If a cubic with real coefficients has one real solution then the circumcentre lies on the Real axis.

Some areas

If and are adjacent sides and the included angle in a parallelogram, the area is given by

If and are adjacent sides and the included angle in a triangle, the area is given by

[kamil9876]

lol i was thinking about some geometry earlier this week. Consider the formula /0 provided to find area of triangle: . Now say we want to find the area of an iscoseles triangle that has side lengths with an angle of in between. The area of this is Alternatively the area can be found by dropping a perpendicular and evaluating , and hence the area is . If you equate the two expressions for area and cancel out the 0.5 you get:


My favourite's: Don't try to trisect an angle with compass and straight edge, waste of time. Don't try to square the circle.