Download Neutral Geometry Theorems Theorem 1. Every line segment has a

Neutral Geometry Theorems
Theorem 1. Every line segment has a midpoint.
Theorem 2. Every angle has a unique angle bisector
Theorem 3. Supplements of congruent angles are congruent.
Theorem 4. Complements of congruent angles are congruent.
Theorem 5. Vertical angles are congruent.
Theorem 6. (Pasch’s Theorem) If a line ` instersects a triangle ∆ABC at a point S with A − S − B
then ` intersects AC or BC.
←→
←→
Theorem 7. Let AB be a line, and suppose that C and D are points not on AB and such that A − C − D.
←→
Then C and D are on same side of the line AB.
Theorem 8. Suppose ∠CAB is an angle, and that P and C are in the same half plane determined by the
←→
line AB. If m(∠P AB) < m(∠CAB) then P is interior to ∠CAB.
Theorem 9. (The Crossbar Theorem) Suppose P is point which is in the interior of an angle ∆CAB.
−→
Then the ray AP intersects BC at a point D with B − D − C.
Theorem 10. (The Isosceles Triangle Theorem) If two sides of a triangle are congruent, the the angles
opposite these two sides are congruent.
Theorem 11. A point is on the perpendicular bisector of a line segment if and only if it is equidistant to
the endpoints of that line segment.
Theorem 12. (The Exterior Angle Theorem) An exterior angle to a triangle has greater measure than
either of the associated remote interior angles of the triangle.
Theorem 13. (The Angle-Side-Angle Congruence Theorem) If the vertices of one triangle are in
one-to-one correspondence with the vertices of a second triangle so that two angles and the included side of
the first triangle are congruent to the corresponding two angles and included side of the second triangle, then
the triangles are congruent.
Theorem 14. (Converse to the Isosceles Triangle Theorem) If two angles of a triangle are congruent,
then the sides opposite these two angles are congruent.
Theorem 15. (Angle-Angle-Side Congruence Theorem) If there is a correspondence between the
vertices of two triangles so that two angles and a non-included side of one triangle are congruent to the
corresponding two angles and the non-included side of the second triangle, then the triangles are congruent.
Theorem 16. (The Hypotenuse-Leg Theorem.) Let ∆ABC and ∆DEF be such that ∠A and ∠D
are right angles, AB ∼
= DE and BC ∼
= EF , then ∆ABC ∼
= ∆DEF .
Theorem 17. Suppose ∆ABC is a triangle, and AB > AC. Then m(∠C) > m(∠B)
Theorem 18. (Triangle Inequality Theorem) Let ∆ABC be a triangle. Then AB + BC > AC.
Theorem 19. (The Hinge Theorem) Suppose there is a one-to-one correspondence between the vertices
of one triangle and the vertices of a second triangle so that two sides of the first triangle are congruent to the
corresponding sides of the second triangle. Suppose the included angle of the first triangle has larger measure
than the corresponding included angle of the second triangle. Then the side opposite the included angle of
the first triangle has larger measure than the side opposite the included angle of the second triangle.
Theorem 20. (Side-Side-Side Congruence Theorem) Suppose the vertices of one triangle are in oneto-one correspondence to the vertices of a second triangle so that corresponding sides are congruent. Then
the two triangles are congruent.
Theorem 21. (Alternating Interior Angle Theorem) If two lines are crossed by a transversal so that
a resulting pair of alternating interior angles are congruent, then the lines are parallel.
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Corollary 22. Suppose two lines are crossed by a transversal. If any of the following result, then the lines
are parallel.
• A resulting pair of corresponding angles are congruent.
• A resulting pair of interior angles on the same side of the transversal are supplementary.
• The transversal is perpendicular to both lines.
Theorem 23. Euclid’s Fifth Postulate is equivalent to Playfair’s Postulate (also called the Euclidean Parallel
Postulate).
Lemma 24. The sum of any two angles of a triangle have angle sum less than 180.
Lemma 25. Suppose ∆ABC is a triangle. Then there is a triangle ∆DEF with the property that the angle
sum of ∆DEF equals the angles sum of ∆ABC and m(∠EDF ) ≤ 12 m(∠ABC).
Theorem 26. (The Sacchieri-Legendre Theorem) The angle sum of any triangle is less than or equal
to 180.
←→
Theorem 27. Given a line ` and a point P not on `, there is a unique point Q on ` such that P Q is
parallel to `. Furthermore, if R is any other point on `, then P Q < P R.
Euclidean Geometry Theorems
Theorem 28. (The Converse of the Alternate Interior Angle Theorem) If two parallel lines are
crossed by a transversal, then alternate interior angles are congruent.
Theorem 29. The sum of the measures of the interior angles of a triangle is 180.
Notation: We denote the sum of the three interior angle measures of a triangle ∆ABC by ΣABC .
Corollary 30. The measure of an exterior angle of a triangle is equal to the sum of the measures of the two
remote interior angles.
Theorem 31. Opposite sides of a parallelogram are congruent.
Theorem 32. If three (or more) parallel lines intercept congruent segments on one transversal t, then they
intercept congruent line segments on any other transversal.
Theorem 33. (The Median Concurrence Theorem.) The three medians of a triangle are concurrent at
a point called the centroid. Furthermore, any two medians of a triangle intersect at a point that is two-thirds
the distance from any vertex to the midpoint of the opposite side.
Theorem 34. Parallelograms that share a common base and that have sides opposite this base contained in
the same line, are equal in area.
Theorem 35. If ` and m are parallel lines and A and B are points on m, then the distance from A to `
equals the distance from B to `.
Theorem 36. The area of a parallelogram is the product of the lengths of its base and its height.
Theorem 37. The area of a triangle is one-half the product of the length of its base and the corresponding
height.
Lemma 38. Let ∆ABC be a triangle. Let ` and m be perpendicular bisectors to AB and BC respectively.
Then ` and m intersect.
Theorem 39. The perpendicular bisectors of the sides of a triangle are concurrent (at a point of concurrence
called the circumcenter. Furthermore, the circumcenter is the center of a circle containing all three vertices
of the triangle called the circumcircle.
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Corollary 40. Given three distinct non-collinear points in a plane, there is a unique circle containing these
points.
Theorem 41. A point is on an angle bisector if and only if it is interior to the angle and equidistant from
the sides of the angle.
−→
−→
Lemma 42. Suppose ∆ABC is a triangle and let AP and BQ be interior angle bisectors. Then these rays
intersect.
Theorem 43. The three bisectors of the interior angles of a triangle are concurrent (at a point called the
incenter.) Furthermore, there is a circle whose center point is the incenter which contains exactly one point
on each side of the triangle.
Theorem 44. (The Basic Proportionality Theorem.) Let ∆ABC be a triangle. Let ` intersect AB
and AC at points other than vertices D and E respectively, The following are equivalent:
←→
i) ` is
AD
ii) DB
AB
iii) AD
parallel to BC .
AE
∼
.
= EC
AC
∼
= AE .
Theorem 45. (The Angle Angle Angle Similarity Theorem - (AAA).) If, under a correspondence
between two triangles, the three interior angles of one triangle are congruent to the corresponding three
interior angles of the other triangle, then the triangles are similar.
Theorem 46. (The Side Angle Side Similarity Theorem - (SAS).) If, under a correspondence between
two triangles, an angle of one triangle is congruent to the corresponding angle of the other triangle, and if
the corresponding sides that surround this angle are proportional, then the triangles are similar.
Theorem 47. (The Side Side Side Similarity Theorem - (SSS).) If, under a correspondence between
two triangles, the lengths of the three sides of one triangle are proportional to the corresponding lengths of
the three sides of the other triangle, then the triangles are similar.
Theorem 48. (The Pythagorean Theorem.) If a and b are the lengths of the sides of a right triangle,
and if c is the length of the hypotenuse, then a2 + b2 = c2 .
Theorem 49. The lines containing the three altitudes of a triangle are concurrent.
Theorem 50. Each angle bisector of a triangle is concurrent with the bisectors of the exterior angles at the
remaining two vertices. The point of concurrence is called the excenter of the triangle. Each excenter of a
triangle is the center of a circle that is tangent to the lines containing the sides of the triangle.
Theorem 51. The orthocenter, the circumcenter and the centroid of a given triangle are collinear (on a
line called the Euler line for the triangle.)
Corollary 52. The distance from any vertex of a triangle to the orthocenter of the triangle is twice the
distance from the circumcenter to the midpoint of the side opposite the vertex.
Theorem 53. (Nine Point Circle.) Let ∆ABC be a triangle. The nine points consisting of
i) the midpoints of the sides of ∆ABC,
ii) the feet of the three altitudes of the triangle,
iii) the midpoints of the segments formed by joining the orthocenter to the vertices of the triangle,
are concyclic. The center of this nine point circle is the midpoint of the segment joining orthocenter and the
circumcenter of ∆ABC, and the length of the radius of this circle is one-half the length of the circumradius
of ∆ABC.
←→
Definition 1. Suppose we have a line AB and points C and D on that line. Place coordinates on that line
using the Ruler Postulate. The signed distance from C to D is the number rC − rD .
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Comments:
• Since the notion of signed distance in only used in the following theorems, and should be clear from
context when it is used, we use the same symbol for signed distance as we do for the usual notion of
distance between points.
• The value of the signed distance between two points depends on how one chooses coordinates for
the line. However, it follows from the Ruler Postulate that the absolute value of the signed distance
is the usual notion of distance between points. Thus, given two points on a line, there are only two
possible values for the signed distance between these two points, and these two values have the same
absolute value.
• If A, B and C are collinear, then the number AB
AC does not depend on how one chooses coordinates. If choosing different coordinates for the line results in the numerator changing sign then the
denominator will do so as well, and the value of the quotient will remain unchanged.
AB
AB
• If A − B − C then AB
AC > 0. If B − A − C then AC < 0. If A − C − B then AC > 0.
Definition 2. Three points are Menelaus points for the triangle ∆ABC if each point lies on a distinct
line determined by the sides of the triangle ∆ABC.
Theorem 54. (Menelaus Theorem.) Suppose ∆ABC is a triangle and X, Y , and Z are Menelaus points
←→ ←→
←→
on AC, BC and AB respectively. Then X, Y and Z are collinear if and only if
BY
CX
AZ
×
×
= −1.
ZB
YC
XA
In the above equation, distance is the signed distance mentioned above.
Definition 3. A Cevian line for a triangle ∆ABC is a line joining a vertex to a point on the opposite
side of the triangle from this vertex.
←→ ←→
←→
Theorem 55. (Ceva’s Theorem.) Suppose AX, BY and CZ are Cevian lines for the triangle ∆ABC
with points X, Y and Z on the sides BC, AC and AB respectively. These lines are concurrent if and only if
BX
CY
AZ
×
×
= 1.
ZB
XC
YA