演绎几何.doc

Chapter 5 More on Deductive Geometry Name:____________( ) Class: ( )
Important Terms
deductive geometry
演繹幾何
altitude
頂垂線(高)
congruent triangle
全等三角形
in-centre
內心
similar triangle
相似三角形
inscribed circle
內切圓
angle bisector
角平分線
circumcentre
外心
perpendicular bisector
垂直平分線
circumcircle
外接圓
median
中線
centroid.
形心
concurrent
共點
orthocenter
垂心
Revision Notes
1. Using the deductive approach to solve geometric problems that involve triangles.
Based on the properties or conditions of congruent triangles, similar triangles and isosceles triangles, we can use deductive approach to prove and obtain more geometric results.
2. Some special lines in a triangle
A
B
C
D
(a) An angle bisector is a line segment which divides an angle into two equal parts.
. In the figure, AD is the angle bisector of .
(b) A perpendicular bisector is a line which bisects a side and is perpendicular to that side.
A
B
C
D
E
. In the figure, DE is the perpendicular bisector of AC.
A
B
C
D
(c) A median is a line segment which joins a vertex to the mid-point of its opposite side.
. In the figure, BD is a median.
(d) An altitude is a perpendicular line segment which drops from a vertex to its opposite side.
C
A
B
D
. In the figure, BD is an altitude.
3. Triangle Inequality
a
c
b
In any triangle, the sum of the lengths of any two sides is always longer than the length of the third side.
. In the figure, a + b > c,
b + c > a,
c + a > b.
4. Relations between the lines in a triangle
A
B
C
P
Q
R
I
(a) The 3 angle bisectors of a triangle are always concurrent.
The point of intersection is called the in-centre.
A circle (inscribed circle) can be constructed with
the in-centre as the centre and touching the 3 sides
of the triangle.
. In the figure, I is the in-centre
and PQR is the inscribed circle.
(b) The 3 perpendicular bisectors of a triangle are always concurrent.
O
B
A
C
The point of inte