**1. Centroid: The centroid of a triangle is the intersection
of the three medians of the triangle. A median is the line that
passes through the mid-point of one side of the triangle and the
opposite vertex. Centroid is generally denoted by G.Click
here for a GSP script for Centroid (G) of a triangle ABC.**

**2. Orthocenter:The orthocenter of a triangle is the intersection
of the three altitudes of the triangle. An altitude is the line
that passes through a vertex and is perpendicular to the opposite
side of the triangle. Orthocenter is generally denoted by H.Click here for a GSP script for Ortocenter
(H) of a triangle ABC.**

**3. Circumcenter:The circumcenter of a triangle is the intersection
of the three perpendicular bisectors of the triangle. A perpendicular
bisectors is the line that passes through the mid-point of one
side of the triangle and is perpendicular to that side. Circumcenter
is generally denoted by C. Click here
for a GSP script for Circumcenter (C) of a triangle ABC.**

**4. Circumcircle: A Triangle's circumscribed circle. Its center O
is called the Circumcenter, and its Radius R the Circumradius. Click here for a GSP script for Circumcircle
of a triangle ABC.**

**5. Incenter: The Incenter of a triangle is the intersection
of the three angle bisectors of the triangle. A angle bisectors
is the line that passes through a vertex an divides that interior
angle of the triangle in half.Click here
for a GSP script for Incenter of a triangle ABC.**

**6. Incircle: The Inscribed Circle of a Triangle . The center
I of the incircle is called the Incenter and the Radius r the
Inradius.Click here for a GSP script
for Incircle of a triangle ABC.**

**7. Medial triangle:The medial triangle is the one whose vertices are
the midpoints of the sides of a given triangle. Click
here for a GSP script for the medial triangle of a triangle
ABC.**

**7a. Orthocenter, Mid-segment
triangle: The midsegment triangle is
the one whose vertices are the midpoints of the segments AH, BH,
and CH (where H is the orthecenter of the triangle ABC). Click here for a GSP script
for the medial triangle of a triangle ABC.**

**8. Orthic triangle:
The orthic triangle is constructed
by connecting the feet of the altitudes of a triangle ABC. Click here for a GSP script for
the medial triangle of a triangle ABC.**

**9. Pedal triangle: The pedal triangle is created by selecting an arbitrary
triangle with an arbitrary point (P) which could be either inside
or outside the triangle.Then the intersection of the three perpendicular
lines from P to the sides of the triangle creates the pedal triangle.
Click here for a GSP script for
a pedal triangle of a triangle ABC.**

**10. Nine Point Circle:
The nine point circle is the circle
that ,in a triangle ABC, the midpoints of AB, BC, and AC; the
points at the feet of the altitudes;and the midpoints of the segments
connecting the vertices of triangle ABC to the orthocenter lye
on. Click here for a GSP script
for the nine-point circle.**

**11. Center of Nine point
circle:The center of the nine point
circle lies on Euler's Line midway between the circumcenter and
the orthocenter. Click here for
a GSP script for the center (N) of a nine-point circle.**

**12. Trisecting a line
segment:Click
here for a GSP script for trisecting a given line segment
AB.**

**13. Equilateral triangle,
given a side: Click
here for a GSP script for the construction of the equilateral
triangle, given a side AB.**

**14. Square, given a
side: Click here
for a GSP script for the construction of the square, given a side
AB.**

**15. Isosceles triangle,
given base and altitude: Click
here for a GSP script for the construction of the equilateral
triangle, given the base AB and altitude.**

**16. Triangle Centers
(H, G, C, and I): Click
here for a GSP script for constructing all four centers (H,
G, C and I) of a triangle ABC.**

**17. Triangle Centers
with Euler Line: In any triangle, three
remarkable points - circumcenter, centroid, and orthocenter -
are colinear, that is, lie on the same line, the Euler line. Centroid
is always located between the circumcenter and the orthocenter
twice as close to the former as to the latter.Click
here for a GSP script for Triangle Centers (H, G, C, and I)
with Euler Line.**

**18. Locus of vertex
of a fixed angle that subtends a fixed segment: Locus
of vertex of a fixed angle that subtends a fixed segment AB is
a circle with radious AB. Click here
for a GSP script for the Locus of vertex of a fixed angle that
subtends a fixed segment AB.**

**19. Divide a segment
AB into two parts that form a golden ratio: The
Golden Ratio () is an irrational number with several curious properties.
It can be defined as that number which is equal to its own reciprocal
plus one: = 1/ + 1.
Multiplying both sides of this same equation by the Golden Ratio
we derive the interesting property that the square of the Golden
Ratio is equal to the simple number itself plus one: : ^2
= + 1. This gives us either 1.618
033 989 or -0.618 033 989. The first number is usually regarded
as the Golden Ratio itself, the second as the negative of its
reciprocal. Click here for a GSP script
for finding a point R on a segment AB s.t. AR/SR=.**

**20. Pentagon, given
a radius: A pentagon is a five sided
regular polygon. Click here for a GSP
script for constructing a pentogon with given the radious of its
circumcircle.**

**21. Pentagon, given a side: Click here for
a GSP script for constructing a pentogon with given a side.**

**22. Hexagon, given a
side: A hexagon is an eight sided regular
polygon. Click here for a GSP script
for constructing a hexagon with given a side.**

**23. Octagon, given a
side: Click here for a GSP script for
constructing an octagon with given a side.**

**24. Rhombus, given a side: A rhombus is a parallelogram with all four sides
equal. Click here for a GSP script
for constructing a rhombus with given a side.**

**25. Inscribed circle of a rhombus
rhombus, given a side: Inscribed circle
is the circle tangent to all four sides of a rhombus. Click
here for a GSP script for inscribing rhombus.**

**26. Square, given the diagonal:Click here for a GSP
script for constructing a square with given its diagonal.**

**27. Fermat point: Fermat Point inside a triangle is the point s.t.
the sum of the distances from this point to three vertices is
the minimum.Click here for a GSP script
for constructinf the Fermat Point in a triangle.**

**28. Inscribing an equilateral triangle: Inscribing equilateral triangle means placing
an quilateral triangle inside a circle. Click
here for a GSP script for inscribing an equailateral triangle
in a circle.**

**28. Hexagon, given radious: Click here for a GSP
script for constructing a hexagon with given radious of its circumcircle.**

**33. Logarithmic Spiral (Equiangular
Spiral, Golden Spiral, Spira Mirabilis):**

Equiangular spiral is a curve that cuts all radii vectors at a constant angle.

Explanation:

1.Let there be a spiral (that is, any curve r==f[theta] where
f is a monotonic function)

2.From any point P on the spiral, draw a line toward the center
of the spiral. (this line is called the radial vector)

3.If the angle formed by the radial vector and the tangent for
any point P is constant, the curve is an equiangular spiral.

Click Here for a GSP script for constructing a logarithmic spiral

__Note:__
After using the script

Drag point D to change the constant angle.

Drag point K to adjust the position of radius vector.

Drag point M to increase/decrease range of curves ploted.

*This page created
September 24, 1999*

*This page last
modified November 17, 1999*