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A FOUNDED METHOD OF NEW THEOREMS OF PLANE
GEOMETRY
Nguyen Ngoc Giang
1. INTRODUCTION
To find out new theorems in general and new geometric theorems
in particular is all of our dream. That’s reason why, researchers give not
only nice and beautiful theorems but also founded methods of new
geometric theorems. These are two important factors of researchers in this
era. In fact, there are a lot of founded methods of new theorems such as
affine-homogeneity method in higher geometry books. This small paper
refers to a founded method of new theorems, that is the method of
generalization from a line to a circle.
2. THE CREATIVE METHOD CONTENT OF NEW GEOMETRIC
THEOREMS
The content is rather simple, the line is the particular case of a
circle when the circle has center lying at infinity. That ‘s reason why we
generalize from segments, lines to arcs, circles to obtain new problems.
3. ILLUSTTRATED EXAMPLES
Theorem 1 (Pascal theorem)
Given 6 vertices ABCDEF
inscribed in a circle. Let
X  AB  DE , Y  AF  DC , Z  BC  EF . Prove that
X , Y, Z
are
collinear.
Theorem 1’ (Pascal theorem ‘s generalization)
Given 6 vertices ABCDEF inscribed in a circle and d is an
arbitrary line. Let Oab be the point of intersection of mid-perpendicular
AB and d. Let (Oab ) be the circle with its center Oab and its radius Oab A .
Similarly to the circles (Ode ), (Obc ), (Oef ), (Oaf ), (Odc ). Let X , X '; Y , Y '; Z , Z '
be the points of intersection of circles (Oab ), (Ode ); (Oaf ), (Ocd ); (Oef ), (Ocb ),
respectively. Prove that 6 points lie on a circle and its center lies on d .
When d is the line at infinity, the circle passing through X , Y , Z is the
Pascal line.
E
C
A
Y
Z
X
F
D
d
B
O
Oab
X'
Y'
Z'
Theorem 2 (Simson ‘s theorem)
Given a triangle ABC inscribed in a circle with center O. P is an
arbitrary point lying on the circle. Drop perpendiculars PA ', PB ', PC '
from P to BC , CA, AB, respectively. Prove that A ', B ', C ' are collinear.
Theorem 2’ (Simson theorem ‘s generalization)
Given a triangle ABC inscribed in a circle with center O. P is an
arbitrary point lying on the circle. d is an arbitrary line. Let Oab be the
point of intersection of the mid-perpendicular of segment AB with d. Let
(Oab ) be the circle with center Oab and radius Oab A . Let (Oab' ) be the circle
passing through P and being orthogonal to (Oab ). Similarly, we have the
pairs of orthogonal circles (Obc ), (Obc' ); (Oca ), (Oca' ) . Three pairs of circles
intersect at 6 points A ', A ''; B ', B ''; C ', C ''. prove that 6 points
A ', A ''; B ', B ''; C ', C '' lie on a conic that d is the symmetric axis of conic.
When d is the line at infinity then conic degenerates to a pair of
lines and then we have Simson’s theorem.
4. AAPLICATED PPROBLEMS
Please to apply the above method to the following problems
Theorem 3 – Dao’s theorem (Simson theorem ‘s generalization)
Given a triangle ABC inscribed in a circle ( O ). P is an arbitrary
point on ( O ). An arbitrary line d passing through O intersects
PA, PB, PC at D, E , F . Let X , Y , Z be the projection from the points
D, E , F to BC , CA, AB . Prove that X , Y , Z lie on the same line.
Theorem 4
Given a triangle ABC and O is arbitrary. Drop perpendiculars
from O to OA, OB, OC and intersect BC , CA, AB at A ', B ', C ' , respectively.
Prove that A ', B ', C ' are collinear.
5. REMARKS
1. The above method has a close relation with Lobachevskian
plane geometry. We discovery the following theorem:
Theorem 5
Prove that in Lobachevskian plane geometry, three medians are
concurrent..
What is the expression of this theorem? We have the comparative
table as follow:
Euclidian look
Lobachevskian look
1. Line 
1. Curve at infinite 
2. Point is on the above side of 
2. Point
3. Point belongs to 
3. Point at infinity
4. Point belongs to the below side of 4. (There ‘s nothing corresponding to this
point)

5. An semicircle belongs to the above
side of  and its center belongs to 
(this center can be a point at infinity
and that time the semicircle becomes
to a ray).
6. A circle arc, which belongs to the
above side of  , is on a circle that its
center belongs to  (This center can
be a point at infinity) and there ‘s an
endpoint belonging to  .
7. Angle is formed by two arcs AU,
AV mentioned in the part 6 (U, V
belong to D )
8. The magnitude of the angle formed
two arcs AU, AV is mentioned in the
part 7 (the angle unit is degree or
radian)
9. Arc AB belongs to a semicircle
being on the above side of  and its
center belongs to  (this center can
be a point at infinity)
10. R | ln(ABUV) | where U, V are
two intersection points of D with the
semicircle mentioned in the part 7 (U
or V is a point at infinity if the
semicircle is a ray) and R is a ratio
coefficient.
11. The geometric transformations
M ® M'
corresponding
with
5. A line
6. A ray
7. Angle is formed by two rays AU, AV (U,
V are points at infinity)
8. The magnitude of the UAV angle is
mentioned in the part 7 (the angle unit is
degree or radian)
9. Segment AB
10. The length of the segment AB.
11. The geometric transformations.
ax + b
ax + b
and x ' =
cx + d
cx + d
( ad - bc ¹ 0 ), where a, b, c, d are
real numbers, x and x’ are complex
numbers (x = u + iv) and M, M’ are
two points corresponding with two
complex numbers x, x’ and
x ' = x = u - iv , is the symmetric
transformation to horizontal axis Ou.
12. The inversion with a pole A (A
12. The symmetric transformation to a line
belongs to D ), power r 2 : is expressed by an Euclidian semicircle with
AM . AM ' = r 2
the center A and the radius r.
x' =
Thus, the poof of theorem 5 is equivalent to the proof of the
median problem extended from the line to the circle.
2. Because the centers of three intersected circles lie on d , there is
a conic passing through the 6 points that d is the symmetric axis.
6. PROPOSAL
- Please to research all of plane geometric problems according to
the method extended from the line to the circle. We will obtain interesting
results as above. We need to extend the basic notions such as “midpoint”,
“bisectors”, “orthogonal”, etc. and then we extend all of theorems of
plane geometry.
REFERENCES
[1] Forum BAI TOAN HAY – LOI GIAI ĐEP – DAM ME TOAN HOC
[2] Nguyen Canh Toan (1992), To practice to good students to make the
acquaintance of researching mathematics, Vietnam Educational
publishing house.
The address : Nguyen Ngoc Giang
Email: [email protected]