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Let N be the nine-point center of a triangle ABC. Let A'B'C' be the circumcevian triangle of N, and let A''B''C'' be the pedal triangle of N with respect to A'B'C'. The Euler lines of A'B''C'', B'C''A'', C'A''B'' concur in X(10227). (Tran Quang Hung and Angel Montesdeoca, September 14, 2016.) See 24387 .

X(10227) lies on these lines: (pending)

a^2 (a^26-8 a^24 (b^2+c^2)+28 a^22 (b^2+c^2)^2-6 a^20 (9 b^6+28 b^4 c^2+28 b^2c^4+9 c^6)+a^18 (53 b^8+277 b^6 c^2+406 b^4 c^4+277 b^2 c^6+53 c^8)+a^16 (6 b^10-273 b^8 c^2-499 b^6 c^4-499 b^4 c^6-273 b^2 c^8+6 c^10)+a^14 (-96 b^12+184 b^10 c^2+307 b^8 c^4+386 b^6 c^6+307 b^4 c^8+184 b^2 c^10-96 c^12)+2 a^12 (66 b^14-56 b^12 c^2-9 b^10 c^4-38 b^8 c^6-38 b^6 c^8-9 b^4 c^10-56 b^2 c^12+66 c^14) -a^10 (69 b^16+6 b^14 c^2+97 b^12 c^4+43 b^10 c^6+5 b^8 c^8+43 b^6 c^10+97 b^4 c^12+6 b^2 c^14+69 c^16)-a^8 (28 b^18-238 b^16 c^2+176 b^14 c^4-11 b^12c^6+63b^10 c^8+63 b^8 c^10-11 b^6 c^12+176 b^4 c^14-238 b^2 c^16+28 c^18)+a^6 (b^2-c^2)^2 (68 b^16-256 b^14 c^2+89 b^12 c^4+65 b^10 c^6+97 b^8 c^8+65 b^6 c^10+89 b^4 c^12-256 b^2 c^14+68 c^16)-a^4 (b^2-c^2)^4 (46 b^14-120 b^12 c^2+26 b^10 c^4+73 b^8 c^6+73 b^6 c^8+26 b^4 c^10-120 b^2 c^12+46 c^14) +a^2 (b^2-c^2)^6 (15 b^12-29 b^10 c^2+16 b^8 c^4+32 b^6 c^6+16 b^4 c^8-29 b^2 c^10+15 c^12)-(b^2-c^2)^8 (2 b^10-3 b^8 c^2+5 b^6 c^4+5 b^4 c^6-3 b^2 c^8+2 c^10) )

Let N be the nine-point center of a triangle ABC. Let A'B'C' be the circumcevian triangle of N, and let A''B''C'' be the pedal triangle of N with respect to A'B'C'. The triangle A''B''C'' is similar to ABC, and the center of similitude is X(10228). (Tran Quang Hung and Angel Montesdeoca, September 14, 2016.) See 24387 .

X(10228) lies on these lines: (pending)

Let OaObOc be the anticevian triangle of X(3). Let Pa = X(3)-of-BCOa, and define Pb and Pc cyclically. Then X(10229) = PaPbPc-to-ABC cyclologic center; see X(10152 for the 2nd Schmidt cyclologic center and Quadri-Figures-Group #1936

X(10229) lies on these lines: (pending)

Let I = X(1). Let A' be the I-excenter of BCI, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(10230). (Randy Hutson, December 10, 2016)

X(10230) lies on these lines: {79,1127}, {174,482}, {483,8351}, {3082,8092}

X(10230) = isogonal conjugate of X(10232)

X(10231) lies on these lines: {1,164}, {43,361}, {168,1743}, {188,8092}, {1128,10023}, {6726,8090}

X(10231) = isogonal conjugate of X(10215) X(10231) = Gibert-Burek-Moses concurrent circles image of X(164) X(10231) = X(1324)-of-intouch triangle X(10231) = {X(1),X(266)}-harmonic conjugate of X(1130)

X(10232) lies on this line: {1,168}

X(10232) = isogonal conjugate of X(10230) X(10232) = {X(1),X(259)}-harmonic conjugate of X(1129)

X(10233) lies on these lines: {1,188}, {164,166}, {165,505}, {167,8140}, {6726,8423}, {8089,10215}

X(10233) = {X(1),X(188)}-harmonic conjugate of X(10234)

The following special string values are permitted (case need not be respected):

A number MAY also have a trailing:

This lexical formatting may be modified using these additional properties:

decimalChar groupChar bareNumber

format : no options (other than the default).

The field contains integers - that is whole numbers.

Integer values are indicated in the standard way for any valid integer.

Additional properties:

The field contains boolean (true/false) data.

In the physical representations of data where boolean values are represented with strings, the values set in trueValues and falseValues are to be cast to their logical representation as booleans. trueValues and falseValues are arrays which can be customised to user need. The default values for these are in the additional properties section below.

The boolean field can be customised with these additional properties:

trueValues falseValues

The field contains data which is valid JSON.

The field contains data that is a valid JSON format arrays.

A date without a time.


A time without a date.

A date with a time.

A calendar year as per XMLSchema gYear .

Usual lexical representation is YYYY . There are no format options.

A specific month in a specific year as per XMLSchema gYearMonth .

Usual lexical representation is: YYYY-MM . There are no format options.

A duration of time.

We follow the definition of XML Schema duration datatype directly and that definition is implicitly inlined here.

To summarize: the lexical representation for duration is the ISO 8601 extended format PnYnMnDTnHnMnS, where nY represents the number of years, nM the number of months, nD the number of days, 'T' is the date/time separator, nH the number of hours, nM the number of minutes and nS the number of seconds. The number of seconds can include decimal digits to arbitrary precision. Date and time elements including their designator may be omitted if their value is zero, and lower order elements may also be omitted for reduced precision.

The field contains data describing a geographic point.


The field contains a JSON object according to GeoJSON or TopoJSON spec.


Any type or format is accepted. When converting from physical to logical representation, the behaviour should be similar to String field type.

A richer, "semantic", description of the "type" of data in a given column MAY be provided using a rdfType property on a field descriptor.

The value of of the rdfType property MUST be the URI of a RDF Class, that is an instance or subclass of RDF Schema Class object

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