IC 4406: A Seemingly Square Nebula and Golden Rectangles

Evidence indicates that IC 4406 is likely a hollow cylinder, with its square appearance the result of our vantage point in viewing the cylinder from the side.
Credit: C. R. O'Dell (Vanderbilt U.) et al., Hubble Heritage Team, NASA.



Read more at:
gogeometry.com/wonder_world/ic_4406_square_nebula_nasa_golden_rectangle.html

Problem 275: Right Triangle, Circumcircle, Sagitta, Inradius

Proposed Problem



See complete Problem 275 at:
gogeometry.com/problem/p275_right_triangle_sagitta_inradius.htm

Level: High School, SAT Prep, College geometry

Sagitta, Arc, Chord

Definition
Given a circle, the sagitta (Latin for arrow) is the perpendicular distance CD from an arc's midpoint (C) to the chord across it.



Read more at:
gogeometry.com/circle/sagitta_chord_arc_arrow.htm

Asteriod Almahata Sitta 15 and Golden Rectangles

Small asteroid 2008 TC3 fell to Earth at dawn on October 7, 2008, tracking through the skies over the Nubian Desert in northern Sudan. That event was remarkable because it was the first time an asteroid was detected in space before crashing into planet Earth's atmosphere.



Read more at:
gogeometry.com/education/almahata_sitta_15_asteroid_golden_rectangle.html

Singapore MRT Line and Golden Rectangles

The North East MRT Line (NEL) is the third Mass Rapid Transit line in Singapore and the world's first fully-underground, automated and driverless rapid transit line.



Read more at:
gogeometry.com/education/singapore_north_east_mrt_line_golden_rectangle.html

Shanghai Metro and Golden Rectangles

The Shanghai Metro is the urban rapid transit system of China's largest city, Shanghai. The system incorporates both subway and light railway lines. It opened in 1995, making Shanghai the third city in Mainland China, after Beijing and Tianjin, to have a subway.



Read more at:
gogeometry.com/education/shanghai_metro_golden_rectangle.html

Cone Nebula and Golden Rectangles

The Cone Nebula is located about 2,600 light-years from Earth in the constellation Monoceros and is so named because of its apparent shape.



Read more at:
gogeometry.com/education/cone_nebula_golden_rectangle_hubble_telescope.html

Euclid's Elements Book I, Proposition 9

To bisect a given angle




Read more at:
gogeometry.com/geometry/euclid_elements_book_i_9_bisect_angle.htm

Problem 274: Isosceles Triangle, 80-80-20, Angles

Proposed Problem
Try to use elementary geometry (Euclid's Elements.)


See complete Problem 274 at:
gogeometry.com/problem/p274_isosceles_triangle_80_80_20_angles.htm

Level: High School, SAT Prep, College geometry

Isosceles Triangle Index

Theorems and Problems about an isosceles triangle.





Read more at:
gogeometry.com/geometry/isosceles_triangle_index_two_equal_sides.html

Bottema's Theorem: Triangle and Squares

Invariance of an isosceles right triangle.
Triangle and Squares with Interactive Geometry Software
Step-by-Step construction, Manipulation, and animation.




Read more at:
gogeometry.com/geometry/bottema_theorem_triangle_square.htm

Problem 273: Triangle, Perpendiculars, Area of Squares

Proposed Problem



See complete Problem 273 at:
gogeometry.com/problem/p273_triangle_perpendicular_squares_areas.htm

Level: High School, SAT Prep, College geometry

Problem 272. Tangent Circles, the Cube of the Common external tangent

Proposed Problem



See complete Problem 272 at:
gogeometry.com/problem/p272_tangent_circles_cube_common_external_tangent.htm

Level: High School, SAT Prep, College geometry

Rhombus, Rhombuses, Rhombi

Index.




Read more at:
gogeometry.com/geometry/rhombus_rhombi_diamond_lozenge_index.htm

Equilic Quadrilateral: Theorem 3. Rhombus

Proposed Problem

In the figure below ABCD is an equilic quadrilateral. Prove that the midpoints E, G, F, and H of the diagonals and the sides BC and AD always determine a rhombus EFGH.



See more at:
gogeometry.com/equilic/equilic_quadrilateral_03.htm

Level: High School, SAT Prep, College geometry

Problem 271. Tangent Circles, the Cube of the Common external tangent

Proposed Problem



See complete Problem 271 at:
gogeometry.com/problem/p271_tangent_circles_cube_common_external_tangent.htm

Level: High School, SAT Prep, College geometry

Heptagons, Theorems and Problems

Index




Read more at:
gogeometry.com/geometry/heptagon_heptagram_seven_index.htm

Level: High School, SAT Prep, College geometry

Heptagon and Heptagrams / Septegrams

There are two regular heptagrams, labeled by Schlafli symbols as {7/2} and {7/3}, with the second number representing the vertex interval step from a regular heptagram, {7/1}.



Read more at:
gogeometry.com/geometry/heptagon_heptagram_seven.htm

Level: High School, SAT Prep, College geometry

Fractal Landscape of Machu Picchu

Machu Picchu is a pre-Columbian Inca site located 8,000 ft above sea level. It is situated on a mountain ridge above the Urubamba Valley in Peru, which is 50 miles northwest of Cuzco and through which the Urubamba River flows. Often referred to as "The Lost City of the Incas", Machu Picchu is one of the most familiar symbols of the Inca Empire.



Read more at:
gogeometry.com/education/machu_picchu_fractal_landscape.html

Level: High School, SAT Prep, College geometry

Problem 270. Tangent Circles, Common external tangent, Fractional exponents

Proposed Problem



See complete Problem 270 at:
gogeometry.com/problem/p270_tangent_circles_fractional_exponent.htm

Level: High School, SAT Prep, College geometry

Circumcenter Index

Theorems and Problems




Read more at:
gogeometry.com/center/circumcenter_triangle_theorems_problems_index.html

Level: High School, SAT Prep, College geometry

Pi Day: March 14

It's time to get irrational. Tomorrow is Pi Day, when mathematicians will gather to celebrate the mystery of science's most famous strange number.

The month (3) and the day (14) give us 3.14 which is pi to two decimal places. That’s enough for teachers around the world to mark it as International Pi Day.



Read more at:
gogeometry.com/geometry/pi_314_irrational_math_constant_index.html

Level: High School, SAT Prep, College geometry

Problem 269. Right Triangle, Altitude, Perpendicular, Projection

Proposed Problem


See complete Problem 269 at:
gogeometry.com/problem/p269_right_triangle_altitude_orthogonal_projection.htm

Level: High School, SAT Prep, College geometry

Problem 268: Right Triangle, Catheti and Altitude

Proposed Problem


See complete Problem 268 at:
gogeometry.com/problem/p268_right_triangle_catheti_altitude_hypotenuse_square.htm

Level: High School, SAT Prep, College geometry

Problem 267: Right Triangle, Product of Catheti

Proposed Problem


See complete Problem 267 at:
gogeometry.com/problem/p267_right_triangle_catheti_product.htm

Level: High School, SAT Prep, College geometry

Problem 266: Right Triangle, Altitude, Geometric Mean

Proposed Problem


See complete Problem 266 at:
gogeometry.com/problem/p266_right_triangle_altitude_geometric_mean.htm

Level: High School, SAT Prep, College geometry

Australia: Math in crisis as teachers go private

Education

Advanced mathematics is disappearing from public school classrooms, leaving students able to learn only basic math, because the few qualified teachers are being snapped up by the private sector. Source: The Australian by Andrew Trounson.



Read more at:
gogeometry.com/world_news_map/australia_maths_in_crisis_as_teachers_go_private.html

Geometry Education Index

Education

In its broad sense, education refers to any act or experience that has a formative effect on the mind, character, or physical ability of an individual...In its technical sense education is the process by which society, through schools, colleges, universities, and other institutions, deliberately transmits its cultural heritage--its accumulated knowledge, values, and skills--from one generation to another. George F. Kneller, Introduction to the Philosophy of Education (New York: John Wiley and Sons, 1971.)



Read more at:
gogeometry.com/education/index.html

Level: High School, SAT Prep, College geometry

Kepler Mission: A search for habitable planets

The $590-million mission, jointly managed by JPL and NASA, will examine a star-rich stretch of sky for a planet where water could exist in liquid form

... Each rectangle indicates the specific region of the sky covered by each CCD element of the Kepler photometer. There are a total of 42 CCD elements in pairs, each pair comprising a square....



Read more at:
gogeometry.com/world_news_map/kepler_nasa_mission_search_habitable_planets.htm

Level: High School, SAT Prep, College geometry

Albert Einstein and the Pythagorean theorem

A preteen Albert Einstein devised a new proof of the Pythagorean theorem

About the Pythagorean theorem, Albert Einstein wrote:

'For example I remember that an uncle told me the Pythagorean theorem before the holy geometry booklet had come into my hands. After much effort I succeeded in "proving" this theorem on the basis of the similarity of triangles ...



Read more at:
gogeometry.com/pythagoras/einstein_geometry_book.htm

Level: High School, SAT Prep, College geometry

Problem 265: Right Triangle, Pythagorean Theorem

Proposed Problem


See complete Problem 265 at:
gogeometry.com/problem/p265_pythagorean_theorem_right_triangle.htm

Level: High School, SAT Prep, College geometry

Problem 264: Right Triangle, Altitude, Leg projection, Hypotenuse, Similarity, Geometric mean

Proposed Problem


See complete Problem 264 at:
gogeometry.com/problem/p264_right_triangle_leg_projection_hypotenuse.htm

Level: High School, SAT Prep, College geometry

Problem 263: Triangle, Median, Altitude, Orthogonal Projection, Sides

Proposed Problem


See complete Problem 263 at:
gogeometry.com/problem/p263_triangle_median_orthogonal_projection.htm

Level: High School, SAT Prep, College geometry

Cassini Images Bizarre Hexagon on Saturn

The Persistent Hexagon

An odd, six-sided, honeycomb-shaped feature circling the entire north pole of Saturn has captured the interest of scientists with NASA's Cassini mission.

See complete Cassini Images Bizarre Hexagon on Saturn at:
The Persistent Hexagon


Spring Reveals Saturn's Hexagon Jet Stream, NASA's Cassini mission

Hexagons - Index

Theorems and Problems



See complete Hexagons Index at:
gogeometry.com/geometry/hexagon_polygon_shape_index_6_sides.html

Level: High School, SAT Prep, College geometry

Problem 262: Regular Hexagon inscribed in a circle, sum of distances

Proposed Problem


See complete Problem 262 at:
gogeometry.com/problem/p262_regular_hexagon_inscribed_circle_distance.htm

Level: High School, SAT Prep, College geometry

Post your solutions or ideas in the comments.

Problem 261: Regular Pentagon inscribed in a circle, sum of distances

Proposed Problem


See complete Problem 261 at:
gogeometry.com/problem/p261_regular_pentagon_inscribed_circle_distance.htm

Level: High School, SAT Prep, College geometry

Post your solutions or ideas in the comments.

Orthocenter Index

Theorems and Problems



See complete Orthocenter Index at:
gogeometry.com/center/orthocenter_theorems_problems_index_software.html

Level: High School, SAT Prep, College geometry

Post your solutions or ideas in the comments.